http://math.eretrandre.org/mybb/ Thu, 20 Nov 2008 08:39:57 +0100 MyBB http://math.eretrandre.org/mybb/showthread.php?tid=33 Thu, 03 Apr 2008 04:31:35 +0200 http://math.eretrandre.org/mybb/showthread.php?tid=33
Now, let's consider another kind of transfinite tree: a pathological kind where the root may have no children yet still has descendents! It is based on reversing the sense of paths in the previous kind of transfinite tree. Let C be a set with the same cardinality as the maximum number of children per node. Again, we associate strings over C with nodes in the tree, and we define a set of strings to be a tree if they satisfy if x is in the tree, then every suffix of x is also in the tree. The parent of a node x is the string obtained by deleting the first symbol in x. A node x is an ancestor of y if y is a suffix of x. The root node is the empty string, since the empty string is always the suffix of any string.

Clearly, for strings of finite length, this definition is identical to the previous definition. But consider the tree where every node is a string of limit ordinal length (still satisfying the suffix requirement, of course: this is possible because the parent of each string is still of limit ordinal length). The root node has no children, but is an ancestor of every node in the tree!

Even stranger is the tree obtained by taking a finite tree and then appending strings of limit ordinal length to it such that the suffix requirement is satisfied. Since none of the finite strings can possibly be a parent of the limit-ordinal-long strings, the tree consists of two kinds of nodes: one kind reachable by a finite path, and another kind reachable only by an infinite path. These two kinds of nodes are "connected" to the root in two completely different ways; the finite nodes have ancestors which are children of the root, but none of the ancestors of the infinite nodes are the root's children. None of the finite nodes are in the same "branch" as an infinite node, because finite strings are never a suffix of a string that's a limit ordinal long.

The root node doesn't have to be the only node with this peculiar property, of course. We could have paths with lengths of some limit ordinal like w+w, so at depth w we have nodes without children, yet with descendents.

It gets stranger than this. Let's say we now construct a tree with transfinite strings of non-limit ordinal length, say of length w+1. Then, the root does have children; but the children nodes have no further children, yet they do have descendents! Of course, the same thing happens for strings of length L+k, where L is some limit ordinal, and k is a non-zero finite number.

In other words, the difference between the original kind of transfinite tree and the inverted transfinite tree is that in the former, the "leap" across the "infinite gap" always occurs at limit ordinal depths, whereas in the latter, we get to choose when this "leap" happens by selecting a suitable value of k. We can even choose k differently for each branch in the tree. In the former case, we can always "leap" to a unique, minimal depth (we land at a node with no parent); whereas in the latter case, while we need to "leap" to access the infinite depth nodes, there is no minimum depth that we can leap to (we always land at a node with a parent).

Hence, in the original type of transfinite tree, the gaps are always "open upwards": at limit ordinal depths, nodes have no parents but do have ancestors. In inverted transfinite treese, the gaps are always "open downwards": at a certain depth, there are no more children, so you must leap downwards to the infinite descendents. These kinds of gaps are "one-way" gaps, where there is an infinite stretch on one side but not on the other. As long as we use ordinals to index node depths, we will always have only one-way gaps.

But it is possible to conceive of even stranger trees where the paths have infinite gaps that are open both ways: where ancestors are separated from descendents by an infinite gap, but every ancestor has a child and every descendent has a parent---yet the two sides never meet. We cannot index node depths with ordinals in this case: we need to use the equivalent of the closure of ordinals under subtraction. (We can obtain such an index set by using a suitable subset of Conway's surreal numbers.)

Is it possible to have a gap that is closed both ways? Such a thing would mean, hypothetically, that at a certain depth k, there is an "infinite" gap to another depth k', but the node at depth k has no children and the node at depth k' has no parent. But then this implies that there is nothing between the two nodes, and so there is no gap, and k'=k+1. So a gap that's closed both ways is equivalent to no gap at all.

All of this leads one to surmise that we can define a gap to be either open upwards (the usual meaning of transfinite path), open downwards (inverted transfinite trees), open both ways (surreal trees?), or closed (i.e., there is no gap). So a path consists of a sequence of nodes delimited by gaps of any of these 4 kinds, and a tree consists of a set of paths such that if P is in the tree, then every subpath of P is also in the tree. We allow any mixture of different kinds of gaps in a path. If a path has only closed gaps (i.e., no gaps), then it's a finite path; otherwise, it's an infinite path.

I'm not sure where I'm going with this, but it sure sounds interesting. :P]]>
Now, let's consider another kind of transfinite tree: a pathological kind where the root may have no children yet still has descendents! It is based on reversing the sense of paths in the previous kind of transfinite tree. Let C be a set with the same cardinality as the maximum number of children per node. Again, we associate strings over C with nodes in the tree, and we define a set of strings to be a tree if they satisfy if x is in the tree, then every suffix of x is also in the tree. The parent of a node x is the string obtained by deleting the first symbol in x. A node x is an ancestor of y if y is a suffix of x. The root node is the empty string, since the empty string is always the suffix of any string.

Clearly, for strings of finite length, this definition is identical to the previous definition. But consider the tree where every node is a string of limit ordinal length (still satisfying the suffix requirement, of course: this is possible because the parent of each string is still of limit ordinal length). The root node has no children, but is an ancestor of every node in the tree!

Even stranger is the tree obtained by taking a finite tree and then appending strings of limit ordinal length to it such that the suffix requirement is satisfied. Since none of the finite strings can possibly be a parent of the limit-ordinal-long strings, the tree consists of two kinds of nodes: one kind reachable by a finite path, and another kind reachable only by an infinite path. These two kinds of nodes are "connected" to the root in two completely different ways; the finite nodes have ancestors which are children of the root, but none of the ancestors of the infinite nodes are the root's children. None of the finite nodes are in the same "branch" as an infinite node, because finite strings are never a suffix of a string that's a limit ordinal long.

The root node doesn't have to be the only node with this peculiar property, of course. We could have paths with lengths of some limit ordinal like w+w, so at depth w we have nodes without children, yet with descendents.

It gets stranger than this. Let's say we now construct a tree with transfinite strings of non-limit ordinal length, say of length w+1. Then, the root does have children; but the children nodes have no further children, yet they do have descendents! Of course, the same thing happens for strings of length L+k, where L is some limit ordinal, and k is a non-zero finite number.

In other words, the difference between the original kind of transfinite tree and the inverted transfinite tree is that in the former, the "leap" across the "infinite gap" always occurs at limit ordinal depths, whereas in the latter, we get to choose when this "leap" happens by selecting a suitable value of k. We can even choose k differently for each branch in the tree. In the former case, we can always "leap" to a unique, minimal depth (we land at a node with no parent); whereas in the latter case, while we need to "leap" to access the infinite depth nodes, there is no minimum depth that we can leap to (we always land at a node with a parent).

Hence, in the original type of transfinite tree, the gaps are always "open upwards": at limit ordinal depths, nodes have no parents but do have ancestors. In inverted transfinite treese, the gaps are always "open downwards": at a certain depth, there are no more children, so you must leap downwards to the infinite descendents. These kinds of gaps are "one-way" gaps, where there is an infinite stretch on one side but not on the other. As long as we use ordinals to index node depths, we will always have only one-way gaps.

But it is possible to conceive of even stranger trees where the paths have infinite gaps that are open both ways: where ancestors are separated from descendents by an infinite gap, but every ancestor has a child and every descendent has a parent---yet the two sides never meet. We cannot index node depths with ordinals in this case: we need to use the equivalent of the closure of ordinals under subtraction. (We can obtain such an index set by using a suitable subset of Conway's surreal numbers.)

Is it possible to have a gap that is closed both ways? Such a thing would mean, hypothetically, that at a certain depth k, there is an "infinite" gap to another depth k', but the node at depth k has no children and the node at depth k' has no parent. But then this implies that there is nothing between the two nodes, and so there is no gap, and k'=k+1. So a gap that's closed both ways is equivalent to no gap at all.

All of this leads one to surmise that we can define a gap to be either open upwards (the usual meaning of transfinite path), open downwards (inverted transfinite trees), open both ways (surreal trees?), or closed (i.e., there is no gap). So a path consists of a sequence of nodes delimited by gaps of any of these 4 kinds, and a tree consists of a set of paths such that if P is in the tree, then every subpath of P is also in the tree. We allow any mixture of different kinds of gaps in a path. If a path has only closed gaps (i.e., no gaps), then it's a finite path; otherwise, it's an infinite path.

I'm not sure where I'm going with this, but it sure sounds interesting. :P]]> http://math.eretrandre.org/mybb/showthread.php?tid=32 Sun, 03 Feb 2008 23:47:31 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=32

– We can build a “Theory of Numbers”, starting from the positive integers and trying to give them a “group” structure via the definition of a binary operation “+”, and a unit/neutral element “0”. We immediately (so to say) discover that, for justifying the group structure, we need to include within the postulates, the definition of “negative integers”, for allowing the inverse operation. If we go ... further, we may proceed defining the “multiplication group” structure around operation “x” and unit element “1” that, automatically, generates the exigency of defining the rational numbers (including the Z set), for allowing the inverse operation. For similar (but ... more complicated) reasons, the introduction of the “^” operation obliges us to define the set of real and complex numbers, even without reaching the “group structure”, for allowing the two inverse operations. In fact, the “^” operation is non-commutative and the “unit element” cannot be found.
– Having said that (in a .... very informal way), we may observe that the implementation of any ... step (rank) in the hyper-operation hierarchy, dramatically increases the set of postulates of the theory, for justifying the return (inverse) operations. This postulate set is probably infinite-countable and Prof. Goedel would have been happy to read this.
– My problem is that I was always convinced that hyper-operation rank s=3 (exponentiation), not only opened the door to complex (including real) numbers, but also opened ... the window to something that we might call “multiple” or “fibrated(?)” numbers. In fact, I am convinced that the square root of 4 is, indeed, a “double number”: -2 and +2. The fact of indicating it by plus/minus-sqrt(4) is orthodox (and perhaps also ... catholic, i.e. universal ;) ) but does not satisfies my ... dirty mind. Moreover, we might write cbrt(8) = 2 and cbrt(-8) = -2, but we also know that we have, in both cases, two other well defined complex numbers, as solutions of this operation. In fact, the square roots of a number are two and the cubic roots of a number are three. Of course, this is very well known (the problem of the multiple complex roots of number 1). But this is often the subject of some superficially-deep specialist considerations, which we may very well leave out of the ... main window, if we limit ourselves to the ... reality.
– Actually, equation x^n - a = 0 is a particular case of a general algebraic equation of n degree. The “General Theorem of Algebra” says that the multiplicity of solutions of an algebraic equation of n degree is n. In other words, the number of solutions of x^n = a must exactly be n. For example, the four solutions of the fourth root of 16 are: +2, +i2, -2, -i2, which we might put as 4th-rt(16) = {2,i2,-2,-i2}.The way of writing it is not relevant. The tricky (but not ... fuzzy!) idea behind is that, if the fourth root of 16 must be a number (one number), well, this number has to be “multiple”, with multiplicity 4. It is not a question of “choosing” (I know, this is the official point of view), there is nothing to be chosen, it’s just like that. But, this point of view implies the definition of multiple (fibrated? matrix?) numbers. Well ..., you know, a Civilization that was able to swallow number “i” can do much more than that. We just have to check if it is worth while!
– Now (and here I risk to become ... crazy), we know that sqrt(4) = 4^(1/2). Then, we might put 4^(1/2) = {-2,2}, with multiplicity 2. But, how about 4^(1/e)? Or, better (worse?) we know that y = 4^x is a very civilized exponential function, with base 4, very “near” to the shape of y = e^x, mother and sister of all of us (analytic, fully derivable and with y = y’ = y”, and so on). Nevertheless (there is always a ... never-the-less), how to represent its complete behaviour, showing its multiplicities for x = 1/n? Or (again), if x is irrational transcendent (for instance, but not exclusively, < 1), what can we do? Ggggghhhrrrrrhhhoooofff ... ! ;-)
– Last but not least, let us consider y = [x->+oo] lim (sin(x)). Of course, it doesn’t exist, it is indeterminate. Nevertheless (ops ...) it must be a value limited between -1 and +1, i.e. something like y = [-1, 1], a ... continuou-multiple (or limited-undetermined number. What a mess, ... in my mind!


Best regards.
Gianfranco]]>

– We can build a “Theory of Numbers”, starting from the positive integers and trying to give them a “group” structure via the definition of a binary operation “+”, and a unit/neutral element “0”. We immediately (so to say) discover that, for justifying the group structure, we need to include within the postulates, the definition of “negative integers”, for allowing the inverse operation. If we go ... further, we may proceed defining the “multiplication group” structure around operation “x” and unit element “1” that, automatically, generates the exigency of defining the rational numbers (including the Z set), for allowing the inverse operation. For similar (but ... more complicated) reasons, the introduction of the “^” operation obliges us to define the set of real and complex numbers, even without reaching the “group structure”, for allowing the two inverse operations. In fact, the “^” operation is non-commutative and the “unit element” cannot be found.
– Having said that (in a .... very informal way), we may observe that the implementation of any ... step (rank) in the hyper-operation hierarchy, dramatically increases the set of postulates of the theory, for justifying the return (inverse) operations. This postulate set is probably infinite-countable and Prof. Goedel would have been happy to read this.
– My problem is that I was always convinced that hyper-operation rank s=3 (exponentiation), not only opened the door to complex (including real) numbers, but also opened ... the window to something that we might call “multiple” or “fibrated(?)” numbers. In fact, I am convinced that the square root of 4 is, indeed, a “double number”: -2 and +2. The fact of indicating it by plus/minus-sqrt(4) is orthodox (and perhaps also ... catholic, i.e. universal ;) ) but does not satisfies my ... dirty mind. Moreover, we might write cbrt(8) = 2 and cbrt(-8) = -2, but we also know that we have, in both cases, two other well defined complex numbers, as solutions of this operation. In fact, the square roots of a number are two and the cubic roots of a number are three. Of course, this is very well known (the problem of the multiple complex roots of number 1). But this is often the subject of some superficially-deep specialist considerations, which we may very well leave out of the ... main window, if we limit ourselves to the ... reality.
– Actually, equation x^n - a = 0 is a particular case of a general algebraic equation of n degree. The “General Theorem of Algebra” says that the multiplicity of solutions of an algebraic equation of n degree is n. In other words, the number of solutions of x^n = a must exactly be n. For example, the four solutions of the fourth root of 16 are: +2, +i2, -2, -i2, which we might put as 4th-rt(16) = {2,i2,-2,-i2}.The way of writing it is not relevant. The tricky (but not ... fuzzy!) idea behind is that, if the fourth root of 16 must be a number (one number), well, this number has to be “multiple”, with multiplicity 4. It is not a question of “choosing” (I know, this is the official point of view), there is nothing to be chosen, it’s just like that. But, this point of view implies the definition of multiple (fibrated? matrix?) numbers. Well ..., you know, a Civilization that was able to swallow number “i” can do much more than that. We just have to check if it is worth while!
– Now (and here I risk to become ... crazy), we know that sqrt(4) = 4^(1/2). Then, we might put 4^(1/2) = {-2,2}, with multiplicity 2. But, how about 4^(1/e)? Or, better (worse?) we know that y = 4^x is a very civilized exponential function, with base 4, very “near” to the shape of y = e^x, mother and sister of all of us (analytic, fully derivable and with y = y’ = y”, and so on). Nevertheless (there is always a ... never-the-less), how to represent its complete behaviour, showing its multiplicities for x = 1/n? Or (again), if x is irrational transcendent (for instance, but not exclusively, < 1), what can we do? Ggggghhhrrrrrhhhoooofff ... ! ;-)
– Last but not least, let us consider y = [x->+oo] lim (sin(x)). Of course, it doesn’t exist, it is indeterminate. Nevertheless (ops ...) it must be a value limited between -1 and +1, i.e. something like y = [-1, 1], a ... continuou-multiple (or limited-undetermined number. What a mess, ... in my mind!


Best regards.
Gianfranco]]> http://math.eretrandre.org/mybb/showthread.php?tid=31 Mon, 28 Jan 2008 21:08:50 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=31
Now, my question, Cantor did define subsequent epsilon ordinals, but where exactly do they lie in the hierarchy of higher ordinal operations? For example, is defined to be the second ordinal satisfying . Does that mean it is equal to w pentated to 3? Or perhaps w pentated to w?

Also, what about the Ackermann function as applied to the ordinals? Does A(w,w) correspond with some ordinal in the Veblen hierarchy?

I wonder what operation the Feferman-Schütte ordinal corresponds with---if we map it back to functions over the natural numbers, it would seem to me that this would correspond with an extremely fast-growing function. I suspect this function would still be computable, since it is still below the Church-Kleene ordinal. I also wonder if the Church-Kleene ordinal corresponds with the Busy Beaver function. (These considerations are part of my attempt to see how close I can get to the Busy Beaver function while still staying within the realm of computability.)]]>
Now, my question, Cantor did define subsequent epsilon ordinals, but where exactly do they lie in the hierarchy of higher ordinal operations? For example, is defined to be the second ordinal satisfying . Does that mean it is equal to w pentated to 3? Or perhaps w pentated to w?

Also, what about the Ackermann function as applied to the ordinals? Does A(w,w) correspond with some ordinal in the Veblen hierarchy?

I wonder what operation the Feferman-Schütte ordinal corresponds with---if we map it back to functions over the natural numbers, it would seem to me that this would correspond with an extremely fast-growing function. I suspect this function would still be computable, since it is still below the Church-Kleene ordinal. I also wonder if the Church-Kleene ordinal corresponds with the Busy Beaver function. (These considerations are part of my attempt to see how close I can get to the Busy Beaver function while still staying within the realm of computability.)]]> http://math.eretrandre.org/mybb/showthread.php?tid=30 Fri, 07 Dec 2007 17:12:55 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=30 http://math.eretrandre.org/mybb/showthread.php?tid=29 Sat, 01 Dec 2007 03:42:23 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=29
The Basic Idea

The idea is simple: instead of restricting ourselves to 1 dimension, where we get a picture that is very difficult to visualize much less understand, we use 2 dimensions instead, by 'wrapping' each stage around to fill 2D, and iterate that into a fractal-like structure.


The Infinite Image

Here's our first crack at it: first, we begin with a unit square. This represents 0, the first finite ordinal. Then we apply the successor function to get 1, which we represent by a rectangle to the right of the unit square, with the same height but half the width. Next, we append to the right of these two a third rectangle, still the same height, but with 1/4 the original width. We take the limit of the process by repeating this, ad infinitum, halving the width of each subsequent rectangle.

This gives us w, the first infinite limit ordinal. It fits in the 2x1 rectangular area (since the half-widths has a limit at 2).

This idea, of course, is not new. Many have used analogous schemes to visualize the transfinite ordinals. The difference comes in the next step. Instead of merely applying the successor function, we make the next step by copying the rectangles corresponding to w, halving their heights, and placing the copies above the originals. Now we have a 2x1.5 rectangular area representing the structure of 2*w. Then we take another copy of w, make its height 1/4 the original, and stack it on 2*w, to get a 2x1.75 rectangular area representing the structure of 3*w.

Now we take the limit, and end up with a 2x2 rectangular area, representing w*w.

Then, we take this 2x2 area, scale it by (1,1/2) so that its height remains the same, but with half its original width, and place this squished copy to the right of the original. So we get a 3x2 area representing . Repeat this and take the limit, and we get a 4x2 area representing .

Then we take the 4x2 area, scale it to half its height, and stack it on itself, and repeat this, halving the height each time, and take the limit, and we get a 4x4 area representing .

The pattern is now clear. At each step, we take the limit of pasting copies of the square area into halved areas, switching between the X and Y axes. Every two steps yields a square area, and so we can repeat this process.

Now, take the limit of this process, and we get an infinitely large square representing . Each rectangle (or square) represents a distinct order type in the ordinal.

The Finite Image

The picture above has the disadvantage of being infinitely large, so it still doesn't give us a full view of the completed set. Is it possible to create a completed view of the set? We shall try.

First, notice that every two steps of the above process yields a square 4 times larger than the original (twice the original's dimensions). This suggests that we can scale the picture after every two steps, so that it will still fit in a unit square area. We can repeat this process any even number of steps, to get a picture of the ordinal corresponding to an ever larger initial segment of . The picture remains within the unit square area.

Finally, we can take the limit as the number of steps go to infinity: we get a fractal-like image, which happens to be the same image we get if we take the limit of repeatedly substituting each rectangle in the 2x2 area for w*w with a suitably scaled version of itself. This final image gives us a view of the completed set . Right...?

Perhaps. Although this interpretation is very tempting, since it finally gives our mind a pictorial way to grasp the full structure of , which is uncountable, it suffers from a few fatal flaws:


Unlike the first, infinite, picture in the previous section, we can no longer identify individual elements in it. The reason for this is that not only have individual rectangles collapsed into dimensionless points, but entire square regions of the finite steps have collapsed into points. In the infinite picture, we still could, if given any subset ordinal, point out what its successor would be. But in this shrunk-down fractal picture, entire stretches of infinite numbers of successors have collapsed into points, and we can no longer distinguish between individual elements of the set, and we can no longer point out what the successor element of an initial segment should be.
This fractal has the property that the lower-left quadrant is identical to a smaller copy of itself. If we continue using the interpretation that a subset of the image should correspond with an ordinal which is an initial segment of the whole, we run into trouble: since the lower-left quadrant is identical in structure to the whole, the 'ordinal' represented by the whole must be a member of itself---a contradiction of the Axiom of Regularity (and indeed, the definition of an ordinal in the first place).


So, sadly to say, although this fractal picture is very compelling, in that it offers us a finite view of the completed set , it cannot possibly correspond with its actual structure. At the most, we can only say that it is an approximate visual aid, to help us grasp its full structure.]]>
The Basic Idea

The idea is simple: instead of restricting ourselves to 1 dimension, where we get a picture that is very difficult to visualize much less understand, we use 2 dimensions instead, by 'wrapping' each stage around to fill 2D, and iterate that into a fractal-like structure.


The Infinite Image

Here's our first crack at it: first, we begin with a unit square. This represents 0, the first finite ordinal. Then we apply the successor function to get 1, which we represent by a rectangle to the right of the unit square, with the same height but half the width. Next, we append to the right of these two a third rectangle, still the same height, but with 1/4 the original width. We take the limit of the process by repeating this, ad infinitum, halving the width of each subsequent rectangle.

This gives us w, the first infinite limit ordinal. It fits in the 2x1 rectangular area (since the half-widths has a limit at 2).

This idea, of course, is not new. Many have used analogous schemes to visualize the transfinite ordinals. The difference comes in the next step. Instead of merely applying the successor function, we make the next step by copying the rectangles corresponding to w, halving their heights, and placing the copies above the originals. Now we have a 2x1.5 rectangular area representing the structure of 2*w. Then we take another copy of w, make its height 1/4 the original, and stack it on 2*w, to get a 2x1.75 rectangular area representing the structure of 3*w.

Now we take the limit, and end up with a 2x2 rectangular area, representing w*w.

Then, we take this 2x2 area, scale it by (1,1/2) so that its height remains the same, but with half its original width, and place this squished copy to the right of the original. So we get a 3x2 area representing . Repeat this and take the limit, and we get a 4x2 area representing .

Then we take the 4x2 area, scale it to half its height, and stack it on itself, and repeat this, halving the height each time, and take the limit, and we get a 4x4 area representing .

The pattern is now clear. At each step, we take the limit of pasting copies of the square area into halved areas, switching between the X and Y axes. Every two steps yields a square area, and so we can repeat this process.

Now, take the limit of this process, and we get an infinitely large square representing . Each rectangle (or square) represents a distinct order type in the ordinal.

The Finite Image

The picture above has the disadvantage of being infinitely large, so it still doesn't give us a full view of the completed set. Is it possible to create a completed view of the set? We shall try.

First, notice that every two steps of the above process yields a square 4 times larger than the original (twice the original's dimensions). This suggests that we can scale the picture after every two steps, so that it will still fit in a unit square area. We can repeat this process any even number of steps, to get a picture of the ordinal corresponding to an ever larger initial segment of . The picture remains within the unit square area.

Finally, we can take the limit as the number of steps go to infinity: we get a fractal-like image, which happens to be the same image we get if we take the limit of repeatedly substituting each rectangle in the 2x2 area for w*w with a suitably scaled version of itself. This final image gives us a view of the completed set . Right...?

Perhaps. Although this interpretation is very tempting, since it finally gives our mind a pictorial way to grasp the full structure of , which is uncountable, it suffers from a few fatal flaws:


Unlike the first, infinite, picture in the previous section, we can no longer identify individual elements in it. The reason for this is that not only have individual rectangles collapsed into dimensionless points, but entire square regions of the finite steps have collapsed into points. In the infinite picture, we still could, if given any subset ordinal, point out what its successor would be. But in this shrunk-down fractal picture, entire stretches of infinite numbers of successors have collapsed into points, and we can no longer distinguish between individual elements of the set, and we can no longer point out what the successor element of an initial segment should be.
This fractal has the property that the lower-left quadrant is identical to a smaller copy of itself. If we continue using the interpretation that a subset of the image should correspond with an ordinal which is an initial segment of the whole, we run into trouble: since the lower-left quadrant is identical in structure to the whole, the 'ordinal' represented by the whole must be a member of itself---a contradiction of the Axiom of Regularity (and indeed, the definition of an ordinal in the first place).


So, sadly to say, although this fractal picture is very compelling, in that it offers us a finite view of the completed set , it cannot possibly correspond with its actual structure. At the most, we can only say that it is an approximate visual aid, to help us grasp its full structure.]]> http://math.eretrandre.org/mybb/showthread.php?tid=28 Mon, 19 Nov 2007 21:50:38 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=28
Transfinite Trees

A finite tree, as we know, consist of a finite set N of nodes, with a distinguished node called the root node, and a set of edges such that there is a unique path from the root node to every other node.

An infinite tree is a tree where the set N is infinite. As has been proven by mathematicians, if every level of an infinite (countable) tree is finite, then there must exists an infinite branch (a branch that extends indefinitely). It is possible to have an infinite tree of finite depth, if some nodes have an infinite number of children. Infinite trees aren't that hard to visualize; they are just trees that extend indefinitely (has one or more infinite branches), or have one or more nodes with an infinite number of children.

But what is a transfinite tree? How do we consistently define nodes that lie beyond finite paths? Here's a possible idea:

Let C be a set of the same cardinality as the maximum number of children per node in the tree. We can then define a finite path as a sequence of elements from C (in other words, a finite path is a string on C), where each element in the sequence represents which child node to traverse next in our path. Each (finitely-reachable) node, n, then corresponds to a unique sequence , which is the path to n. Now, we observe the following property: a node m is an ancestor of n if and only if is a prefix of . In particular, if the length of is precisely one less than the length of , then m is the parent node of n.

So now, we can define a transfinite node n as a node that corresponds with a path of infinite (possibly transfinite) length, satisfying the property that all of its prefixes also lie in the tree. Note that transfinite nodes with paths that correspond with limit ordinals have no parent node, but they do have ancestor nodes!

Since paths and nodes have a 1-to-1 correspondence, we can define transfinite trees simply as a collection of (possibly transfinite) strings on C, satisfying the property that if a string S is in the tree, then every prefix of S is also in the tree. The null string then corresponds with the root node, strings which are not prefixes of other strings in the tree are leaf nodes, and the remaining strings are internal nodes.

Note that transfinite trees have the property that even if C is finite, the resulting tree could be uncountable. For example, let C={0,1}, and take all sequences of C of length L. If L is finite, the corresponding tree is countable (indeed, finite). But if L is , then the tree is infinite, because its leaf nodes corresponds with infinite sequences of C, which are uncountably many!

Quine Atoms and other Pathological Sets

How does all of this apply to Quine Atoms? Well, given a set A with elements that are themselves sets, we may consider A as the root node of a tree, and the elements of A as nodes under A, and the elements of those elements as nodes under these nodes, and so forth. In other words, the tree represents the "containment hierarchy" of A. The well-founded sets of ZFC then correspond with trees of finite depth. If we allow infinite trees, we see that the Quine Atom Q corresponds with an infinite linear sequence of nodes, since Q has itself as its sole element.

Now, if we generalize this to transfinite trees, we suddenly realize that Quine Atoms need not be unique; because it is possible for two sets Q and R, satisfying Q={Q} and R={R} respectively, to differ only at the transfinite depth in their containment hierarchies. For example, the tree corresponding to Q may have no leaf node, and the tree corresponding to R may have a transfinite leaf node at depth . In set-theoretic notation, Q={{{{...}}}} and R={{{{... {} ...}}}}. Thus, both sets satisfy the Quine Atom definition, but they differ at the transfinite depth: Q is "bottomless" whereas an empty set lies at the "bottom" of R.

In fact, we can substitute the transfinite node at depth with any arbitrary tree, possibly transfinite in themselves, and thus generate an infinitude of different sets all satisfying the Quine Atom definition.

If we consider arbitrary transfinite trees, we find many Quine-Atom-like sets that do not necessarily contain themselves as such, but may contain sub-elements that are not reachable via finite applications of the 'element-of' relation. I.e., they are not well-founded sets. However, none of these sets are truly pathological sets such as the one in the Russell Paradox.

The interesting thing about such infinitely-deep contained nodes is that they are not reachable by finite applications of the 'element-of' relation, and so their exact structure at transfinite depth is of limited utility. However, the fact that they differ in spite of behaving identically at finite depths allows us to use them as distinct urelements. We are also safe from running into Russell's Paradox.]]>
Transfinite Trees

A finite tree, as we know, consist of a finite set N of nodes, with a distinguished node called the root node, and a set of edges such that there is a unique path from the root node to every other node.

An infinite tree is a tree where the set N is infinite. As has been proven by mathematicians, if every level of an infinite (countable) tree is finite, then there must exists an infinite branch (a branch that extends indefinitely). It is possible to have an infinite tree of finite depth, if some nodes have an infinite number of children. Infinite trees aren't that hard to visualize; they are just trees that extend indefinitely (has one or more infinite branches), or have one or more nodes with an infinite number of children.

But what is a transfinite tree? How do we consistently define nodes that lie beyond finite paths? Here's a possible idea:

Let C be a set of the same cardinality as the maximum number of children per node in the tree. We can then define a finite path as a sequence of elements from C (in other words, a finite path is a string on C), where each element in the sequence represents which child node to traverse next in our path. Each (finitely-reachable) node, n, then corresponds to a unique sequence , which is the path to n. Now, we observe the following property: a node m is an ancestor of n if and only if is a prefix of . In particular, if the length of is precisely one less than the length of , then m is the parent node of n.

So now, we can define a transfinite node n as a node that corresponds with a path of infinite (possibly transfinite) length, satisfying the property that all of its prefixes also lie in the tree. Note that transfinite nodes with paths that correspond with limit ordinals have no parent node, but they do have ancestor nodes!

Since paths and nodes have a 1-to-1 correspondence, we can define transfinite trees simply as a collection of (possibly transfinite) strings on C, satisfying the property that if a string S is in the tree, then every prefix of S is also in the tree. The null string then corresponds with the root node, strings which are not prefixes of other strings in the tree are leaf nodes, and the remaining strings are internal nodes.

Note that transfinite trees have the property that even if C is finite, the resulting tree could be uncountable. For example, let C={0,1}, and take all sequences of C of length L. If L is finite, the corresponding tree is countable (indeed, finite). But if L is , then the tree is infinite, because its leaf nodes corresponds with infinite sequences of C, which are uncountably many!

Quine Atoms and other Pathological Sets

How does all of this apply to Quine Atoms? Well, given a set A with elements that are themselves sets, we may consider A as the root node of a tree, and the elements of A as nodes under A, and the elements of those elements as nodes under these nodes, and so forth. In other words, the tree represents the "containment hierarchy" of A. The well-founded sets of ZFC then correspond with trees of finite depth. If we allow infinite trees, we see that the Quine Atom Q corresponds with an infinite linear sequence of nodes, since Q has itself as its sole element.

Now, if we generalize this to transfinite trees, we suddenly realize that Quine Atoms need not be unique; because it is possible for two sets Q and R, satisfying Q={Q} and R={R} respectively, to differ only at the transfinite depth in their containment hierarchies. For example, the tree corresponding to Q may have no leaf node, and the tree corresponding to R may have a transfinite leaf node at depth . In set-theoretic notation, Q={{{{...}}}} and R={{{{... {} ...}}}}. Thus, both sets satisfy the Quine Atom definition, but they differ at the transfinite depth: Q is "bottomless" whereas an empty set lies at the "bottom" of R.

In fact, we can substitute the transfinite node at depth with any arbitrary tree, possibly transfinite in themselves, and thus generate an infinitude of different sets all satisfying the Quine Atom definition.

If we consider arbitrary transfinite trees, we find many Quine-Atom-like sets that do not necessarily contain themselves as such, but may contain sub-elements that are not reachable via finite applications of the 'element-of' relation. I.e., they are not well-founded sets. However, none of these sets are truly pathological sets such as the one in the Russell Paradox.

The interesting thing about such infinitely-deep contained nodes is that they are not reachable by finite applications of the 'element-of' relation, and so their exact structure at transfinite depth is of limited utility. However, the fact that they differ in spite of behaving identically at finite depths allows us to use them as distinct urelements. We are also safe from running into Russell's Paradox.]]> http://math.eretrandre.org/mybb/showthread.php?tid=27 Tue, 21 Aug 2007 11:47:03 +0200 http://math.eretrandre.org/mybb/showthread.php?tid=27 Regards,
Rosy.
---------------------------
https://www.esumz.com]]> Regards,
Rosy.
---------------------------
https://www.esumz.com]]> http://math.eretrandre.org/mybb/showthread.php?tid=25 Wed, 27 Jun 2007 03:31:13 +0200 http://math.eretrandre.org/mybb/showthread.php?tid=25
                                       By

                           R. Stephen Newberry


ABSTRACT:  Deduction vs. Induction ~ Logicism  vs. Empiricism ~ Determinism vs. Randomicity/ Chaos, and the
general topic of Contingency are discussed.

Preamble:

One of the more pleasant aspects of advanced age (I’m seventy-nine) is having the leisure in which to renew old literary acquaintances, and thus to savor again the riches of past encounters.  A short list would include such treasures as The Bible (Old Testament, Torah), Shakespeare, the great English essayists of the 17th, 18th, and 19th centuries, and the great logicians of the late 19th and early 20th centuries.  Here, I’ll certainly mention Dedekind, Cantor, Frege, Russell, Skolem, G\”odel, Gentzen, Herbrand, Church, and (God Bless him!) Stephen Cole Kleene, whose Mathematical Logic (Dover edition of 1967) I am now reading again, with a pleasure that verges upon joy.

I had, for now well over forty years, peripatetically been searching for an explanation of the “Cusp of Anomaly“ that exists between the Deductive and the Inductive methods of investigation, (this study in itself would properly be subsumed under the rubric of methodology.)

By just under forty years ago I had gathered “all the bits and pieces” to hand, but stubbornly (mulelishly?) kept getting confused between the apparently disjoint phenomena of  \mathbb{\omega}-inconsistency, non-standard model-theory, the epistemological entailments of  deductive vs. inductive reasoning,  Logicism vs. Empiricism,  determinacy/time-reversibility vs. randomness/dissipative phenomena, and the like.  I was particularly fascinated by the failure of almost all of the 20th century logicians to discuss the role of contingency in classical logic.  (Satisfiability and contingency are not synonymous, although the latter does entail the former.)  
About five years ago, it began to dawn upon me (finally!)  that the goal of my quest was to be found only in the synthesis of these several viewpoints, (“pictures” as Wittgenstein would have put it), and that they were all merely different aspects of this same “Cusp of Anomaly” that had bedeviled my contemplations over so many decades.

Imagine then, my present delight in discovering that Kleene had already put it together, and done it so smoothly and so elegantly, that over the many previous readings, I’d simply missed the point, and had continued to put myself through the utterly unnecessary purgatory of reconstructing the entire edifice ab initio, (including a truly hellish period of “diagonal-crankery”.)

The balance of this essay is an attempt to present “The Cusp” in terms sufficiently unfamiliar that “the picture” may be seen from a perspective that some might find novel, even though it most probably will be, for most readers, a very old story indeed.

My first encounter with “The Cusp” was in the winter of 1961.  I was reviewing my competence with high-school algebra and trigonometry as preparation for the SAT exam, with the intention of  “going back to school” at CCNY and getting at least a BS in EE.  I was at that time 33 years of age, and painfully aware that I had missed my opportunity by not going into the Navy and getting into the “V2” program when I’d had the chance, (which would have led to the same end which now I was intent upon pursuing).   [I had instead chosen the Merchant Marine, as had my father and my grandfather, and for myself it was a very bad choice.]

The review process had been going well enough that, in order to keep myself entertained I was also reading some other books on mathematics, primarily of the popularization genre, among which was George Gamow’s “One, Two, Three, . . . , Infinity”.   I’d already read Russell’s  An Introduction to Mathematical Philosophy, knew something of the Dedekind approach to the foundations of analysis, and had developed a very pretty mental model of the real line.  My model was countable, since at that time I  had no reason to think otherwise, and already had learned that both the rationals and the algebraic irrationals were countable, and since clearly, there was only one remaining block of the partition, the transcendentals, then the transcendentals must certainly be countable, since they have to “fit-in-between” the rationals and the algebraic irrationals.  (Countably-many rational/algebraic-irrationals entails only countably-many places where transcendentals can fit!  Hence, the concept of the continuum.)

On encountering Gamow’s presentation of Cantor’s “Diagonal Proof” of the uncountability of the reals I was deeply affronted and offended:  Gamow’s explanation that the “transcendentals were denser on the real line” than the rationals and the algebraic irrationals was patently hogwash.  It took me not more than perhaps five or ten minutes to come up with a constructive refutation, based upon the fact that, given any two transcendentals, taken arbitrarily close together, one can very easily construct a rational interpolant, and having once constructed that initial rational interpolant, then arbitrarily many subsequent rational interpolants may just as easily be constructed between the lower of the two transcendentals and the initial rational interpolant; and then again arbitrarily many subsequent rational interpolants between the upper of the two transcendentals and the initial rational interpolant; and then again arbitrarily many subsequent rational interpolants between all the previous rational interpolants, again and again, ad infinitum.  So the concept of “denser-on-the-line” just doesn’t work, and without it, neither does the uncountability of the reals.  QED.  This is the first glimpse I had of the “The Cusp”.

So began an indescribably unpleasant twenty years’ bout with that particular form of mental-illness known psychiatrically as obsessionalism, or perseverance, and in the mathematical community as “Diagonal Crank”-ism.  Not one of my many grad-student friends could find fault with the rational-interpolant construction, but no matter, it was clearly a matter of unchallengeable mathematical faith that the “Diagonal Proof” did in fact demonstrate the greater transfinite cardinality of the transcendentals.  (It was on the same metaphysical plane as that of the “immaculate-conception” for devout Roman Catholics.)  Case closed.

Later, I encountered the “Diagonal” construction occurring in the G\”odel proof of 1931, and proofs of the non-recursive-enumerability of the recursively-definable functions, and several other interesting cases,  that simply defied any attempt at refutation, and hence must be accepted as true. The “Cusp” gets ‘curiouser and curiouser’.  HOW  to reconcile the “Cusp”?  

It seemed pretty clear that there was some sort of epistemological affinity between recursive non-enumerability,  and the “non-denumerability of the continuum”, (despite the obvious “apples-and-oranges” objection),  and I was beginning to suspect that it might sometimes be possible to prove-by-induction propositions that were not universally-valid, and hence not syntactically/deductively provable.  The “Cusp” again.  G\”odel proved the deductive completeness (semi-completeness) of FOL, and the so-it-then-seemed deductive incompleteness of the Simple Theory of Types*; then Skolem, and subsequently, Henkin proved the equivalence of  (many-sorted) FOL with  STT,  and the completeness of both FOL and STT, (which makes sense), but then Second order Logic, (which I had previously thought to be a subset of  STT) is not even semi-decidable.  The “Cusp” had me thoroughly confused. [Of course, the “might-sometimes” conjecture was precisely the content of the first G\’odel Incompleteness Theorem, but it took me several re-readings to be able to see that, and even then I was uncertain for a time.]

*  (Already an error, since G\”odel’s “language P” is just STT + PA, and it is PA which is contingent.  But that didn’t really “turn on the light-bulb” until much later.)


I’d succeeded in accepting Tarski’s (Hilbert’s?) \mathbb{\omega}-rule without too much difficulty, because I’d already learned (from L\”owenheim) about the existence of  n-valid propositions, (“fleeing equations” in his parlance) but the fact that the \mathbb{\omega}-rule induced \mathbb{\omega}-inconsistency when adjoined to the axioms of standard number-theory made some warning buzzers go off in my head, and it seemed that non-standard model-theory might be somehow involved there; but all of the non-standard models that I had actually MET were really weird, essentially involving a universe which contains things like “infinite-integers” and infinitesimals, and suchlike.  (Hmmm. . . )  But!
              NEED THAT NECESSARILY BE TRUE OF ALL
non-standard-models?  If a finite set of propositions is non-contradictory then it must have a countable model  (L\”owenheim again), and if that set of propositions is not true in the Standard Model, then clearly that model must be non-standard; as, indeed,  Henkin says that all of the non-full models in his universe of General Models must be; and, by-the-way, these non-standard, non-full models are all countable, which is fine by me, but how come the full general model is standard and non-countable???

As long as Henkin is leaving out some second-order entities, his general models are non-standard.  All he requires of his general models is that the axioms and rules of inference are true in the model, and that the space of entities be closed under Boolean operations.  Any Boolean Lattice satisfies those conditions.  Suppose we had a way of constructing a Boolean Lattice in such manner that all-but-only the non-predicatively-definable sets were omitted:  That would constitute a (non-full) non-standard General Model, and since I’ve never met (or
even heard of) a predicatively-definable uncountable set . . .    THIS might be a good approach to resolving the the “Cusp” problem!!  (And so it proved to be, but I’m getting way ahead of my story . . . )

The CCNY plan fell through.  My SATs were fine but my wife died (malignant melanoma, unutterably awful), and although I then had an income of $50 per week, even in those days that was not enough to live off of in New York City, and I knew that I would be unable to maintain a full academic load in E.E. at CCNY,  and at the same time earn a living.   (In the vain attempt to “drown my grief”, I was also at that time drinking a bottle of Scotch every three days.)  A good friend persuaded the Columbia Physics Department to give me a job that would keep me sober for at least 8 or 10 hours a day, as a technician on a low-energy physics experiment then being carried out at Brookhaven National Laboratories, and it did quite a bit better than that.  

BNL had a research library that kept on the shelves, among other goodies, the entire Mathematical Foundations list of  the North-Holland Publishing Company, and a full back-issue file of the Journal for Symbolic Logic, so that, not only did I have to stay sober while on the job, but for a good several hours thereafter, reading in the library.  And at the end of that,  I did not really need a drink in order to get to sleep!  

It was also my first encounter with a Xerox machine, and no limit was placed on how much one could use it, so I copied reams of good stuff from the Reviews sections of JSL, as well as several volumes of the North-Holland list, that I could in no way understand at that time, but hoped to understand in future.  Those Xeroxes traveled with me to Europe, and lived with me for several years in Lund, Sweden, while I was studying at Matematiska Institut, vid Lunds Universitet,  (at which I was never able formally to register, but nonetheless was permitted to attend lectures and take exams.)

Mathematical Swedish doesn’t really have all that many new words, but Swedish academic standards were very high, math is a demanding subject, and I was simply too exhausted from attending lectures in a still–foreign-language to be able to take good notes, and then to work all the exercises, so I didn’t really make a success of it. (It was in Sweden that I first learned to have examination anxiety!).  Several years later, another friend, an American physicist who was then a visiting consultant at the Lunds Physiska Institut, made some remarks about G\”odels Theorem, which he had encountered at Stanford, and I gently corrected him, explaining that such-and-such was not really the point, but that so-and-so, and he responded,  “Man, you are wasting your time here!”, which was already beginning to become apparent to me.  That led to more discussion of what I was still hoping to do with my life, and in consequence of all this, he persuaded me to go back to the US, go to California, attend  “one of the easier campuses of the UC or Cal State systems”, and apply for student aid.

Fine:  After four years of studying and learning Swedish and still failing my maths exams, it didn’t take too awfully long to persuade me; but then he began to get enthusiastic about the whole thing, and nothing would do but that I should  also apply to Stanford University!  I said, “WHAAAAT?!?!, how am I going to afford that on fifty dollars a week, when I couldn’t make it at CCNY in  NY?”  He said, “No problem, they have lots of money at Stanford.  If you can get admitted, and survive the freshman year, then they won’t let you fail to continue merely for financial problems.”  So I applied, and he coached me on how to go at it, and wrote a letter in my behalf to the head of the Philosophy Department (I’d given up on the EE, but the department was giving Philosophy credit for some Computer Science courses, and that seemed like a better choice), and in the end, I was admitted.  And I did survive the freshman year, but Stanford’s generosity did not materialize (by that time I was thirty-nine years old), and for the first two years I lived in a garage on bread and peanut-butter for breakfast, no lunch, and a half-can of catfood with a handful of rice for dinner.  Fortunately I had volunteered to grade papers in the CS101 course, and Bill McKeeman (the prof) was so pleased with the job I did  that he gave me a job as an R.A. in the Computer Science department, which helped to pay for the catfood.  When I attained Junior status I got a Federal Loan that took me through the rest of the way.  But the Xeroxes had made the trip back with me, and I had kept on reading Logic.  I graduated with a  BA degree in Philosophy, having by that time taken, (or at least attended lectures for) almost all the courses in Computer Science, and as well, attended some of the seminars in Logical Foundations of Mathematics conducted by Sol Feferman and George Kreisel.  

Sol Feferman (may he live in good health), had accepted the task of being my senior advisor while I was still in my ‘Diagonal Crank’ phase.  (He didn’t know that at the time, because I’d learned to keep quiet about it, but by the time that I had finally produced my senior thesis in an acceptable form, he’d realized the mess I was in.)  It was only through his virtually super-human patience that I was finally disabused of my compulsive delusion.  My relief (BEING a Diagonal Crank is no bed of roses!) was exceeded only by my chagrin and contrition, and I entered that time of life known as “the mid-life crisis”, a two-and-a-half-year episode of acute clinical depression.

In the ensuing nearly thirty years,  I caused him no further bother, as I had sworn off of Pure Mathematics, and, in one the most colossally naïve acts of a life characterized by naïve acts, I  gave my entire mathematical library (excepting only Church’s Introduction, Kleene’s Metamathematics, the Woodger edition of Tarski’s Logic, Semantics, Metamathematics, van Heijenoort’s source book, and the Collected Works of Gerhard Gentzen) to Stanford Library on the naively specious assumption that “I can always go back and access them through the Stanford Library System.”   Hah!  No such luck.  My precious library, painfully acquired over a period of more than twelve years, vanished into some black hole, never again to emerge.  (During that time the expense of buying books ensured that I had no difficulty in maintaining the same weight and waist measurement that I’d had at age 23.)

In the meantime, I  did some tech-writing for money, and went back to my other love, music and the Classical Spanish Guitar, while I devoured food, drank an occasional glass of beer or red-wine, and put on fifty pounds and twenty inches of waistline in the process.   I also was fortunate enough to marry another Stanford graduate, a really nice lady who is still my wife, and life was more or less pleasantly normal after that.

Then, one day, as I was innocently walking by my Big Bookshelf, a long, thin, almost membranous, arm reached out from Van’s sourcebook, grabbed me by the ankle, and refused to let go until I took down the book.  I was hooked again.  No longer possessed by the Diagonal Crank obsession, I was now able to accept, grasp, and even to assimilate ideas which for so many years had somehow triggered the D.C. syndrome, and thereupon had immediately become utterly opaque to me.  My infatuation with Predicativity returned; and slowly, painfully, (book prices had increased by an order-of-magnitude), I reacquired most of my mathematics library, and the Don Quixote of Mathematical Philosophy was saddled up again.  Heigh Ho, Rosinante!

I occasionally emailed Sol with questions of a technical nature, and he, always the mensch, answered briefly and promptly.  [“Are you the same ‘Steve Newberry’  I knew thirty years ago at Stanford?”]   In time, I produced an essay which, with much  difficulty, and after over a year of whining and waiting, I finally persuaded him reluctantly to consent to read.

The essay constructively demonstrated (a one–line recurrence, followed by the union of the well-ordered, monotone, transitive, unbounded sequence of finite Boolean Lattices reified by the recurrence) the existence of a non-standard counter-model to Cantor’s Theorem.  Unfortunately, I had neglected adequately to emphasize the non-standard nature of the model, and he, quite understandably, (although still the mensch, still the gentleman) became perturbed, and asked me no longer to address my epiphanic visions to his attention, on grounds of a limited life-expectancy.  As I am five months older than he, and am still loaded with guilt from having already wasted as much or more of his time than he had been accustomed to expend on advising doctoral candidates with their dissertations, I had no option, but to apologize, butt-out and more or less shut up.  


Perhaps it is time to explain the Cusp:  It arises as the critical and essential difference between deductive reasoning [Logicism] and inductive reasoning [Empiricism].  I like to refer to this as the Empirical Vacuity of deductive reason.  Wittgenstein had commented on this at some length in his doctoral thesis Logico Tractatus,  (as had Kant some several centuries earlier.).  In particular, truths derived from purely empirical data cannot be materially implied by tautologous truths from purely logical/syntactic means, because they are in essence empirically vacuous.  [That would be “getting something for nothing”, creating new empirical information from no empirical information:  Not allowed.]

My essay had sought to resolve the “Cusp” by noting that, while inductively-true propositions (n-valid, valid on all-but-only-finite domains) validly imply deductively-true (u-valid, universally valid, tautological) propositions, the converse does not hold.  The propositions which are only-inductively-true are contingent  (Kleene’s word;  Carnap’s term is L-indeterminate), and therefore by definition, necessarily admit of counter-models, which exist only on completed-infinite domains.*  The only theories devoid of finite sub-models that are recognized by the Standard Model of Arithmetic are those “elementarily equivalent with/to” Peano Arithmetic, (which essentially defines the Standard Model of Arithmetic), and hence these other (non-elementarily equivalent) theories with no finite sub-models are categorized as being non-standard, and hence, from the perspective of the SMA and classical Set theory, have no legitimate significance.  

*  That idea is worth a  moment’s reflective thought

Cantor’s theorem, which is true in the SMA, is (only) inductively-true.  We know this because it materially-implies other empirically contingent propositions, e.g., the assertion  of the existence of infinite sets (non-denumerably many!), hence cannot itself be deductively-true, u-valid.  (The “Diagonal” argument is just the recursive construction of an implicitly inductive  [“every” = “all”] proof  of the falsity of a certain class of propositions.   It took me forty years to see that.)

Therefore, Cantor’s theorem must, of necessity, have at least one non-standard counter-model, and the (union of the sequence of) Boolean Lattices generated by the unit-sets of  \mathbb{\omega}  is it.  [It consists of the finite and co-finite sub-sets of  \mathbb{\omega}, augmented by the limit-sets (unions over all the elementary chains in the ultra-filter); of which chains there are obviously only countably many, (since such chains are initiated by only countably-many unit-sets and terminate in only countably-many limit-sets.)

This is a non-standard model, a very pretty model, of all-but-only the Predicatively-definable subsets of  \mathbb{\omega} [on the assumption that unions of countable, well-ordered, monotone, transitive, unbounded sequences are predicative, which Sol several years previously had assured me that “even Poincare would have accepted as predicative”.  (verbatim quote!)]        


So now we have a robustly predicative countable model of  the powerset of \mathbb{\omega} and the countable continuum of predicatively-definable real numbers will be defined by Cauchy-sequences of rationals (which themselves are predicatively-definable within the model), but the original Diagonal-proof of the “uncountability” of the set of all binary sequences is still possible!  Since ALL  entities definable in this system of  STT, are countable, (and let’s agree, from here on, to call the system  ‘PTT’, for “Predicative Theory of Types”, and take it as known that these entities are all-but-only those which  ARE  predicatively-definable) how do we interpret the significance of this construction of a member of  2^\mathbb{\omega}  which is obviously not a member of the sequence from it has just been defined?

Clearly, there must be an alternative interpretation, but what can it be?  

Now it is time to think about the set of all Chaitin/Martin-Lof/Solovay  \mathbb{\OMEGA}-random binary sequences.  [citation]  This is, by definition, necessarily a subset of   2^\mathbb{\omega}  and that subset is, (also by definition)  quintessently  random, and therefore incapable of being well-ordered.  

So the Diagonal-proof demonstrates that there is a (countable?) counter-model to the Axiom of Choice, which is hardly surprising, considering the independence proofs of Cohen and Feferman.  AxCh  is, together with the uncountability of the Cardinality of the Continuum, quintessently contingent, and that is simply a FACT.


“Everything is a trade-off”:  In this case one gives up the non-denumerable continuum in exchange for a denumerable model of the continuum which is categorical, (as are all models of theories expressed in the language PTT), and which omits all-and-only the C/M-L/S \mathbb{\OMEGA}randomly generated binary sequences in  2^\mathbb{\omega}(which may plausibly be conjectured to be the characteristic functions of the uncountable subsets of \mathbb{\omega}.)

This denumerable continuum is known among Pure Mathematicians concerned with Foundational Studies as that which is “provable within  Feferman’s system IR of predicative analysis”, or, equivalently, within “the first-order part of ATR0, and/or ATR0set ”, as set forth in section  VII.3  of  Simpson’s  SUBSYSTEMS, and properly includes the  H-sets  of  section  VIII.3  of  Simpson’s  SUBSYSTEMS, and the Baire space  NN and the Axiom of Countability, together with much of the work of G\”odel developing  the relation between constructible sets and the projective hierarchy.

It also includes a very elegant theory of  recursive transfinite ordinals up to the limit ordinal \mathbb{\GAMMA}0, as developed (independently) by both Solomon Feferman and Kurt Sch\”utte.  Within this formalism can be expressed much of  “measure theory, separable Banach space theory, Ramsey theory, matching theory, well quasiordering theory, and countable algebra”. [cite Simpson SUBSYSTEMS, p. 412]


But what about those ‘uncountable sets’ after all?  Since it is consistent to assert their existence, they must be possible, and therefore must correspond to some real (i.e., existing) entities/phenomena.

I submit that the correspondence is to be found in the FACT that all sets are either capable of being (simply/well) ordered or not.  The former are countable, and the latter are not.  This is a question, not primarily of cardinality, but of structure.  The disjunction is tautologous, and in any context not dependant upon contingent/empirical truth (“Goldbach’s Conjecture, Riemann’s Hypothesis, Twin-Primes Conjecture, &c.)  the issue is determinate, and the principle of excluded-middle applies.


While we’re at it, we might as well dispose of the putative difference between ‘potentially’ and ‘actually’ infinite sets, which has had philosophers at-knives-drawn as far back as Aristotle.  We need merely to stipulate that any (actually-infinite) set  M which is capable of being (simply or well) ordered  is represented by the unbounded (potentially-infinite) sequence  S  which contains all-but-only those elements which appear in  M, and that  *M =df= S;  and where  ‘U’  denotes “union”,  U(S)  =  M;  Here,  ‘*’,  ‘U ‘ are inverse operators.   Thus,  where  ‘X’ may denote the aforementioned   M,  

           *U(X) = U*(X) = U(*X) =  U(S); and U(S)  =  X.

That having been said, it follows very easily that  “M  is uncountably infinite iff  S is unbounded (potentially-infinite) and  \mathbb{\OMEGA}-randomly-generated”,  and  “M  is countably infinite iff  S  is unbounded (potentially-infinite) and orderly-generated”, where ‘orderly-generated ‘  means any one of  “simply-ordered”, “well-ordered” or “recursively-generated”.
[NB:  Cauchy-sequences of rationals are highly non-random.]  But  2^\mathbb{\omega} must of necessity contain all the  C/M-L/S \mathbb{\OMEGA}-randomly-generated infinite binary-sequences that exist!

But what does this do to Set Theory?!?  We’ve been taught that  NBG  is predicative, and there certainly are those who maintain that  ZFC, since formalized entirely within  FOL, must therefore automatically be seen as being predicative.  But PTT demonstrates that at least one version of predicativity simply ignores the existence of the entire subject-matter of classical Set theory!

Well, what do the Masters:  Skolem, Bernays, G\”odel, Kleene et. al.,  have to say on the matter?  They are essentially unanimous that  FOL  is inadequate for the expression of all truths, either of  Set Theory, or of mathematics.  That, (“the Relativity of Set Theory”), is their explanation for the resolution of the L\”owenheim-Skolem “Paradox”, and it works just as well here too.  Different views of the Universe are possible, and therefore consistent.


All done, neat and tidy.


        THAT IS THE “CUSP OF ANOMALY”, and its  RESOLUTION

[Honesty compels me to admit that the correspondence between induction/deduction and standard/non-standard models was not quite as plainly drawn in the essay  as it is here.]


Which brings us back to Kleene’s  Mathematical Logic:  Section 53, Skolem’s Paradox and non-standard models of arithmetic says it all.  On all of my (many!) previous readings of this section, (and so many others like it), I had kept wondering, “Why are these people so obsessed with Peano Arithmetic?  It is as though it were their entire universe!”  [You think?  Really?  Hello? Four years as a Philosophy major at Stanford University, why didn’t I get that sooner?]



I’m tempted to say,  “Now I can die in Peace.”  But it’s not true.  In a life which has been consistently characterized  by naïve enthusiasms,  philosophical revelations, epiphanic insights,  and hypo-manic head-trips,  I  have never more than now  wished to live, at least a little while longer  . . .  maybe get back to the guitar . . .  perhaps try learning to play the violin-cello . . . ,  
                                  
                  but The Cusp has been laid to rest, amen.

Los Altos, CA, 2007]]>
                                       By

                           R. Stephen Newberry


ABSTRACT:  Deduction vs. Induction ~ Logicism  vs. Empiricism ~ Determinism vs. Randomicity/ Chaos, and the
general topic of Contingency are discussed.

Preamble:

One of the more pleasant aspects of advanced age (I’m seventy-nine) is having the leisure in which to renew old literary acquaintances, and thus to savor again the riches of past encounters.  A short list would include such treasures as The Bible (Old Testament, Torah), Shakespeare, the great English essayists of the 17th, 18th, and 19th centuries, and the great logicians of the late 19th and early 20th centuries.  Here, I’ll certainly mention Dedekind, Cantor, Frege, Russell, Skolem, G\”odel, Gentzen, Herbrand, Church, and (God Bless him!) Stephen Cole Kleene, whose Mathematical Logic (Dover edition of 1967) I am now reading again, with a pleasure that verges upon joy.

I had, for now well over forty years, peripatetically been searching for an explanation of the “Cusp of Anomaly“ that exists between the Deductive and the Inductive methods of investigation, (this study in itself would properly be subsumed under the rubric of methodology.)

By just under forty years ago I had gathered “all the bits and pieces” to hand, but stubbornly (mulelishly?) kept getting confused between the apparently disjoint phenomena of  \mathbb{\omega}-inconsistency, non-standard model-theory, the epistemological entailments of  deductive vs. inductive reasoning,  Logicism vs. Empiricism,  determinacy/time-reversibility vs. randomness/dissipative phenomena, and the like.  I was particularly fascinated by the failure of almost all of the 20th century logicians to discuss the role of contingency in classical logic.  (Satisfiability and contingency are not synonymous, although the latter does entail the former.)  
About five years ago, it began to dawn upon me (finally!)  that the goal of my quest was to be found only in the synthesis of these several viewpoints, (“pictures” as Wittgenstein would have put it), and that they were all merely different aspects of this same “Cusp of Anomaly” that had bedeviled my contemplations over so many decades.

Imagine then, my present delight in discovering that Kleene had already put it together, and done it so smoothly and so elegantly, that over the many previous readings, I’d simply missed the point, and had continued to put myself through the utterly unnecessary purgatory of reconstructing the entire edifice ab initio, (including a truly hellish period of “diagonal-crankery”.)

The balance of this essay is an attempt to present “The Cusp” in terms sufficiently unfamiliar that “the picture” may be seen from a perspective that some might find novel, even though it most probably will be, for most readers, a very old story indeed.

My first encounter with “The Cusp” was in the winter of 1961.  I was reviewing my competence with high-school algebra and trigonometry as preparation for the SAT exam, with the intention of  “going back to school” at CCNY and getting at least a BS in EE.  I was at that time 33 years of age, and painfully aware that I had missed my opportunity by not going into the Navy and getting into the “V2” program when I’d had the chance, (which would have led to the same end which now I was intent upon pursuing).   [I had instead chosen the Merchant Marine, as had my father and my grandfather, and for myself it was a very bad choice.]

The review process had been going well enough that, in order to keep myself entertained I was also reading some other books on mathematics, primarily of the popularization genre, among which was George Gamow’s “One, Two, Three, . . . , Infinity”.   I’d already read Russell’s  An Introduction to Mathematical Philosophy, knew something of the Dedekind approach to the foundations of analysis, and had developed a very pretty mental model of the real line.  My model was countable, since at that time I  had no reason to think otherwise, and already had learned that both the rationals and the algebraic irrationals were countable, and since clearly, there was only one remaining block of the partition, the transcendentals, then the transcendentals must certainly be countable, since they have to “fit-in-between” the rationals and the algebraic irrationals.  (Countably-many rational/algebraic-irrationals entails only countably-many places where transcendentals can fit!  Hence, the concept of the continuum.)

On encountering Gamow’s presentation of Cantor’s “Diagonal Proof” of the uncountability of the reals I was deeply affronted and offended:  Gamow’s explanation that the “transcendentals were denser on the real line” than the rationals and the algebraic irrationals was patently hogwash.  It took me not more than perhaps five or ten minutes to come up with a constructive refutation, based upon the fact that, given any two transcendentals, taken arbitrarily close together, one can very easily construct a rational interpolant, and having once constructed that initial rational interpolant, then arbitrarily many subsequent rational interpolants may just as easily be constructed between the lower of the two transcendentals and the initial rational interpolant; and then again arbitrarily many subsequent rational interpolants between the upper of the two transcendentals and the initial rational interpolant; and then again arbitrarily many subsequent rational interpolants between all the previous rational interpolants, again and again, ad infinitum.  So the concept of “denser-on-the-line” just doesn’t work, and without it, neither does the uncountability of the reals.  QED.  This is the first glimpse I had of the “The Cusp”.

So began an indescribably unpleasant twenty years’ bout with that particular form of mental-illness known psychiatrically as obsessionalism, or perseverance, and in the mathematical community as “Diagonal Crank”-ism.  Not one of my many grad-student friends could find fault with the rational-interpolant construction, but no matter, it was clearly a matter of unchallengeable mathematical faith that the “Diagonal Proof” did in fact demonstrate the greater transfinite cardinality of the transcendentals.  (It was on the same metaphysical plane as that of the “immaculate-conception” for devout Roman Catholics.)  Case closed.

Later, I encountered the “Diagonal” construction occurring in the G\”odel proof of 1931, and proofs of the non-recursive-enumerability of the recursively-definable functions, and several other interesting cases,  that simply defied any attempt at refutation, and hence must be accepted as true. The “Cusp” gets ‘curiouser and curiouser’.  HOW  to reconcile the “Cusp”?  

It seemed pretty clear that there was some sort of epistemological affinity between recursive non-enumerability,  and the “non-denumerability of the continuum”, (despite the obvious “apples-and-oranges” objection),  and I was beginning to suspect that it might sometimes be possible to prove-by-induction propositions that were not universally-valid, and hence not syntactically/deductively provable.  The “Cusp” again.  G\”odel proved the deductive completeness (semi-completeness) of FOL, and the so-it-then-seemed deductive incompleteness of the Simple Theory of Types*; then Skolem, and subsequently, Henkin proved the equivalence of  (many-sorted) FOL with  STT,  and the completeness of both FOL and STT, (which makes sense), but then Second order Logic, (which I had previously thought to be a subset of  STT) is not even semi-decidable.  The “Cusp” had me thoroughly confused. [Of course, the “might-sometimes” conjecture was precisely the content of the first G\’odel Incompleteness Theorem, but it took me several re-readings to be able to see that, and even then I was uncertain for a time.]

*  (Already an error, since G\”odel’s “language P” is just STT + PA, and it is PA which is contingent.  But that didn’t really “turn on the light-bulb” until much later.)


I’d succeeded in accepting Tarski’s (Hilbert’s?) \mathbb{\omega}-rule without too much difficulty, because I’d already learned (from L\”owenheim) about the existence of  n-valid propositions, (“fleeing equations” in his parlance) but the fact that the \mathbb{\omega}-rule induced \mathbb{\omega}-inconsistency when adjoined to the axioms of standard number-theory made some warning buzzers go off in my head, and it seemed that non-standard model-theory might be somehow involved there; but all of the non-standard models that I had actually MET were really weird, essentially involving a universe which contains things like “infinite-integers” and infinitesimals, and suchlike.  (Hmmm. . . )  But!
              NEED THAT NECESSARILY BE TRUE OF ALL
non-standard-models?  If a finite set of propositions is non-contradictory then it must have a countable model  (L\”owenheim again), and if that set of propositions is not true in the Standard Model, then clearly that model must be non-standard; as, indeed,  Henkin says that all of the non-full models in his universe of General Models must be; and, by-the-way, these non-standard, non-full models are all countable, which is fine by me, but how come the full general model is standard and non-countable???

As long as Henkin is leaving out some second-order entities, his general models are non-standard.  All he requires of his general models is that the axioms and rules of inference are true in the model, and that the space of entities be closed under Boolean operations.  Any Boolean Lattice satisfies those conditions.  Suppose we had a way of constructing a Boolean Lattice in such manner that all-but-only the non-predicatively-definable sets were omitted:  That would constitute a (non-full) non-standard General Model, and since I’ve never met (or
even heard of) a predicatively-definable uncountable set . . .    THIS might be a good approach to resolving the the “Cusp” problem!!  (And so it proved to be, but I’m getting way ahead of my story . . . )

The CCNY plan fell through.  My SATs were fine but my wife died (malignant melanoma, unutterably awful), and although I then had an income of $50 per week, even in those days that was not enough to live off of in New York City, and I knew that I would be unable to maintain a full academic load in E.E. at CCNY,  and at the same time earn a living.   (In the vain attempt to “drown my grief”, I was also at that time drinking a bottle of Scotch every three days.)  A good friend persuaded the Columbia Physics Department to give me a job that would keep me sober for at least 8 or 10 hours a day, as a technician on a low-energy physics experiment then being carried out at Brookhaven National Laboratories, and it did quite a bit better than that.  

BNL had a research library that kept on the shelves, among other goodies, the entire Mathematical Foundations list of  the North-Holland Publishing Company, and a full back-issue file of the Journal for Symbolic Logic, so that, not only did I have to stay sober while on the job, but for a good several hours thereafter, reading in the library.  And at the end of that,  I did not really need a drink in order to get to sleep!  

It was also my first encounter with a Xerox machine, and no limit was placed on how much one could use it, so I copied reams of good stuff from the Reviews sections of JSL, as well as several volumes of the North-Holland list, that I could in no way understand at that time, but hoped to understand in future.  Those Xeroxes traveled with me to Europe, and lived with me for several years in Lund, Sweden, while I was studying at Matematiska Institut, vid Lunds Universitet,  (at which I was never able formally to register, but nonetheless was permitted to attend lectures and take exams.)

Mathematical Swedish doesn’t really have all that many new words, but Swedish academic standards were very high, math is a demanding subject, and I was simply too exhausted from attending lectures in a still–foreign-language to be able to take good notes, and then to work all the exercises, so I didn’t really make a success of it. (It was in Sweden that I first learned to have examination anxiety!).  Several years later, another friend, an American physicist who was then a visiting consultant at the Lunds Physiska Institut, made some remarks about G\”odels Theorem, which he had encountered at Stanford, and I gently corrected him, explaining that such-and-such was not really the point, but that so-and-so, and he responded,  “Man, you are wasting your time here!”, which was already beginning to become apparent to me.  That led to more discussion of what I was still hoping to do with my life, and in consequence of all this, he persuaded me to go back to the US, go to California, attend  “one of the easier campuses of the UC or Cal State systems”, and apply for student aid.

Fine:  After four years of studying and learning Swedish and still failing my maths exams, it didn’t take too awfully long to persuade me; but then he began to get enthusiastic about the whole thing, and nothing would do but that I should  also apply to Stanford University!  I said, “WHAAAAT?!?!, how am I going to afford that on fifty dollars a week, when I couldn’t make it at CCNY in  NY?”  He said, “No problem, they have lots of money at Stanford.  If you can get admitted, and survive the freshman year, then they won’t let you fail to continue merely for financial problems.”  So I applied, and he coached me on how to go at it, and wrote a letter in my behalf to the head of the Philosophy Department (I’d given up on the EE, but the department was giving Philosophy credit for some Computer Science courses, and that seemed like a better choice), and in the end, I was admitted.  And I did survive the freshman year, but Stanford’s generosity did not materialize (by that time I was thirty-nine years old), and for the first two years I lived in a garage on bread and peanut-butter for breakfast, no lunch, and a half-can of catfood with a handful of rice for dinner.  Fortunately I had volunteered to grade papers in the CS101 course, and Bill McKeeman (the prof) was so pleased with the job I did  that he gave me a job as an R.A. in the Computer Science department, which helped to pay for the catfood.  When I attained Junior status I got a Federal Loan that took me through the rest of the way.  But the Xeroxes had made the trip back with me, and I had kept on reading Logic.  I graduated with a  BA degree in Philosophy, having by that time taken, (or at least attended lectures for) almost all the courses in Computer Science, and as well, attended some of the seminars in Logical Foundations of Mathematics conducted by Sol Feferman and George Kreisel.  

Sol Feferman (may he live in good health), had accepted the task of being my senior advisor while I was still in my ‘Diagonal Crank’ phase.  (He didn’t know that at the time, because I’d learned to keep quiet about it, but by the time that I had finally produced my senior thesis in an acceptable form, he’d realized the mess I was in.)  It was only through his virtually super-human patience that I was finally disabused of my compulsive delusion.  My relief (BEING a Diagonal Crank is no bed of roses!) was exceeded only by my chagrin and contrition, and I entered that time of life known as “the mid-life crisis”, a two-and-a-half-year episode of acute clinical depression.

In the ensuing nearly thirty years,  I caused him no further bother, as I had sworn off of Pure Mathematics, and, in one the most colossally naïve acts of a life characterized by naïve acts, I  gave my entire mathematical library (excepting only Church’s Introduction, Kleene’s Metamathematics, the Woodger edition of Tarski’s Logic, Semantics, Metamathematics, van Heijenoort’s source book, and the Collected Works of Gerhard Gentzen) to Stanford Library on the naively specious assumption that “I can always go back and access them through the Stanford Library System.”   Hah!  No such luck.  My precious library, painfully acquired over a period of more than twelve years, vanished into some black hole, never again to emerge.  (During that time the expense of buying books ensured that I had no difficulty in maintaining the same weight and waist measurement that I’d had at age 23.)

In the meantime, I  did some tech-writing for money, and went back to my other love, music and the Classical Spanish Guitar, while I devoured food, drank an occasional glass of beer or red-wine, and put on fifty pounds and twenty inches of waistline in the process.   I also was fortunate enough to marry another Stanford graduate, a really nice lady who is still my wife, and life was more or less pleasantly normal after that.

Then, one day, as I was innocently walking by my Big Bookshelf, a long, thin, almost membranous, arm reached out from Van’s sourcebook, grabbed me by the ankle, and refused to let go until I took down the book.  I was hooked again.  No longer possessed by the Diagonal Crank obsession, I was now able to accept, grasp, and even to assimilate ideas which for so many years had somehow triggered the D.C. syndrome, and thereupon had immediately become utterly opaque to me.  My infatuation with Predicativity returned; and slowly, painfully, (book prices had increased by an order-of-magnitude), I reacquired most of my mathematics library, and the Don Quixote of Mathematical Philosophy was saddled up again.  Heigh Ho, Rosinante!

I occasionally emailed Sol with questions of a technical nature, and he, always the mensch, answered briefly and promptly.  [“Are you the same ‘Steve Newberry’  I knew thirty years ago at Stanford?”]   In time, I produced an essay which, with much  difficulty, and after over a year of whining and waiting, I finally persuaded him reluctantly to consent to read.

The essay constructively demonstrated (a one–line recurrence, followed by the union of the well-ordered, monotone, transitive, unbounded sequence of finite Boolean Lattices reified by the recurrence) the existence of a non-standard counter-model to Cantor’s Theorem.  Unfortunately, I had neglected adequately to emphasize the non-standard nature of the model, and he, quite understandably, (although still the mensch, still the gentleman) became perturbed, and asked me no longer to address my epiphanic visions to his attention, on grounds of a limited life-expectancy.  As I am five months older than he, and am still loaded with guilt from having already wasted as much or more of his time than he had been accustomed to expend on advising doctoral candidates with their dissertations, I had no option, but to apologize, butt-out and more or less shut up.  


Perhaps it is time to explain the Cusp:  It arises as the critical and essential difference between deductive reasoning [Logicism] and inductive reasoning [Empiricism].  I like to refer to this as the Empirical Vacuity of deductive reason.  Wittgenstein had commented on this at some length in his doctoral thesis Logico Tractatus,  (as had Kant some several centuries earlier.).  In particular, truths derived from purely empirical data cannot be materially implied by tautologous truths from purely logical/syntactic means, because they are in essence empirically vacuous.  [That would be “getting something for nothing”, creating new empirical information from no empirical information:  Not allowed.]

My essay had sought to resolve the “Cusp” by noting that, while inductively-true propositions (n-valid, valid on all-but-only-finite domains) validly imply deductively-true (u-valid, universally valid, tautological) propositions, the converse does not hold.  The propositions which are only-inductively-true are contingent  (Kleene’s word;  Carnap’s term is L-indeterminate), and therefore by definition, necessarily admit of counter-models, which exist only on completed-infinite domains.*  The only theories devoid of finite sub-models that are recognized by the Standard Model of Arithmetic are those “elementarily equivalent with/to” Peano Arithmetic, (which essentially defines the Standard Model of Arithmetic), and hence these other (non-elementarily equivalent) theories with no finite sub-models are categorized as being non-standard, and hence, from the perspective of the SMA and classical Set theory, have no legitimate significance.  

*  That idea is worth a  moment’s reflective thought

Cantor’s theorem, which is true in the SMA, is (only) inductively-true.  We know this because it materially-implies other empirically contingent propositions, e.g., the assertion  of the existence of infinite sets (non-denumerably many!), hence cannot itself be deductively-true, u-valid.  (The “Diagonal” argument is just the recursive construction of an implicitly inductive  [“every” = “all”] proof  of the falsity of a certain class of propositions.   It took me forty years to see that.)

Therefore, Cantor’s theorem must, of necessity, have at least one non-standard counter-model, and the (union of the sequence of) Boolean Lattices generated by the unit-sets of  \mathbb{\omega}  is it.  [It consists of the finite and co-finite sub-sets of  \mathbb{\omega}, augmented by the limit-sets (unions over all the elementary chains in the ultra-filter); of which chains there are obviously only countably many, (since such chains are initiated by only countably-many unit-sets and terminate in only countably-many limit-sets.)

This is a non-standard model, a very pretty model, of all-but-only the Predicatively-definable subsets of  \mathbb{\omega} [on the assumption that unions of countable, well-ordered, monotone, transitive, unbounded sequences are predicative, which Sol several years previously had assured me that “even Poincare would have accepted as predicative”.  (verbatim quote!)]        


So now we have a robustly predicative countable model of  the powerset of \mathbb{\omega} and the countable continuum of predicatively-definable real numbers will be defined by Cauchy-sequences of rationals (which themselves are predicatively-definable within the model), but the original Diagonal-proof of the “uncountability” of the set of all binary sequences is still possible!  Since ALL  entities definable in this system of  STT, are countable, (and let’s agree, from here on, to call the system  ‘PTT’, for “Predicative Theory of Types”, and take it as known that these entities are all-but-only those which  ARE  predicatively-definable) how do we interpret the significance of this construction of a member of  2^\mathbb{\omega}  which is obviously not a member of the sequence from it has just been defined?

Clearly, there must be an alternative interpretation, but what can it be?  

Now it is time to think about the set of all Chaitin/Martin-Lof/Solovay  \mathbb{\OMEGA}-random binary sequences.  [citation]  This is, by definition, necessarily a subset of   2^\mathbb{\omega}  and that subset is, (also by definition)  quintessently  random, and therefore incapable of being well-ordered.  

So the Diagonal-proof demonstrates that there is a (countable?) counter-model to the Axiom of Choice, which is hardly surprising, considering the independence proofs of Cohen and Feferman.  AxCh  is, together with the uncountability of the Cardinality of the Continuum, quintessently contingent, and that is simply a FACT.


“Everything is a trade-off”:  In this case one gives up the non-denumerable continuum in exchange for a denumerable model of the continuum which is categorical, (as are all models of theories expressed in the language PTT), and which omits all-and-only the C/M-L/S \mathbb{\OMEGA}randomly generated binary sequences in  2^\mathbb{\omega}(which may plausibly be conjectured to be the characteristic functions of the uncountable subsets of \mathbb{\omega}.)

This denumerable continuum is known among Pure Mathematicians concerned with Foundational Studies as that which is “provable within  Feferman’s system IR of predicative analysis”, or, equivalently, within “the first-order part of ATR0, and/or ATR0set ”, as set forth in section  VII.3  of  Simpson’s  SUBSYSTEMS, and properly includes the  H-sets  of  section  VIII.3  of  Simpson’s  SUBSYSTEMS, and the Baire space  NN and the Axiom of Countability, together with much of the work of G\”odel developing  the relation between constructible sets and the projective hierarchy.

It also includes a very elegant theory of  recursive transfinite ordinals up to the limit ordinal \mathbb{\GAMMA}0, as developed (independently) by both Solomon Feferman and Kurt Sch\”utte.  Within this formalism can be expressed much of  “measure theory, separable Banach space theory, Ramsey theory, matching theory, well quasiordering theory, and countable algebra”. [cite Simpson SUBSYSTEMS, p. 412]


But what about those ‘uncountable sets’ after all?  Since it is consistent to assert their existence, they must be possible, and therefore must correspond to some real (i.e., existing) entities/phenomena.

I submit that the correspondence is to be found in the FACT that all sets are either capable of being (simply/well) ordered or not.  The former are countable, and the latter are not.  This is a question, not primarily of cardinality, but of structure.  The disjunction is tautologous, and in any context not dependant upon contingent/empirical truth (“Goldbach’s Conjecture, Riemann’s Hypothesis, Twin-Primes Conjecture, &c.)  the issue is determinate, and the principle of excluded-middle applies.


While we’re at it, we might as well dispose of the putative difference between ‘potentially’ and ‘actually’ infinite sets, which has had philosophers at-knives-drawn as far back as Aristotle.  We need merely to stipulate that any (actually-infinite) set  M which is capable of being (simply or well) ordered  is represented by the unbounded (potentially-infinite) sequence  S  which contains all-but-only those elements which appear in  M, and that  *M =df= S;  and where  ‘U’  denotes “union”,  U(S)  =  M;  Here,  ‘*’,  ‘U ‘ are inverse operators.   Thus,  where  ‘X’ may denote the aforementioned   M,  

           *U(X) = U*(X) = U(*X) =  U(S); and U(S)  =  X.

That having been said, it follows very easily that  “M  is uncountably infinite iff  S is unbounded (potentially-infinite) and  \mathbb{\OMEGA}-randomly-generated”,  and  “M  is countably infinite iff  S  is unbounded (potentially-infinite) and orderly-generated”, where ‘orderly-generated ‘  means any one of  “simply-ordered”, “well-ordered” or “recursively-generated”.
[NB:  Cauchy-sequences of rationals are highly non-random.]  But  2^\mathbb{\omega} must of necessity contain all the  C/M-L/S \mathbb{\OMEGA}-randomly-generated infinite binary-sequences that exist!

But what does this do to Set Theory?!?  We’ve been taught that  NBG  is predicative, and there certainly are those who maintain that  ZFC, since formalized entirely within  FOL, must therefore automatically be seen as being predicative.  But PTT demonstrates that at least one version of predicativity simply ignores the existence of the entire subject-matter of classical Set theory!

Well, what do the Masters:  Skolem, Bernays, G\”odel, Kleene et. al.,  have to say on the matter?  They are essentially unanimous that  FOL  is inadequate for the expression of all truths, either of  Set Theory, or of mathematics.  That, (“the Relativity of Set Theory”), is their explanation for the resolution of the L\”owenheim-Skolem “Paradox”, and it works just as well here too.  Different views of the Universe are possible, and therefore consistent.


All done, neat and tidy.


        THAT IS THE “CUSP OF ANOMALY”, and its  RESOLUTION

[Honesty compels me to admit that the correspondence between induction/deduction and standard/non-standard models was not quite as plainly drawn in the essay  as it is here.]


Which brings us back to Kleene’s  Mathematical Logic:  Section 53, Skolem’s Paradox and non-standard models of arithmetic says it all.  On all of my (many!) previous readings of this section, (and so many others like it), I had kept wondering, “Why are these people so obsessed with Peano Arithmetic?  It is as though it were their entire universe!”  [You think?  Really?  Hello? Four years as a Philosophy major at Stanford University, why didn’t I get that sooner?]



I’m tempted to say,  “Now I can die in Peace.”  But it’s not true.  In a life which has been consistently characterized  by naïve enthusiasms,  philosophical revelations, epiphanic insights,  and hypo-manic head-trips,  I  have never more than now  wished to live, at least a little while longer  . . .  maybe get back to the guitar . . .  perhaps try learning to play the violin-cello . . . ,  
                                  
                  but The Cusp has been laid to rest, amen.

Los Altos, CA, 2007]]> http://math.eretrandre.org/mybb/showthread.php?tid=18 Fri, 11 May 2007 08:05:12 +0200 http://math.eretrandre.org/mybb/showthread.php?tid=18
Anyway, I was experimenting with iterated algorithms and stumbled across the following observation, which led me to conjecture that perhaps it's generalizable to n-dimensions, and possibly to any kind of tiling of n-space. However, I'm not clear how to rigorously define my conjecture, so I'm posting here to see what people say.

First, let me explain the "intuitive" motivation behind the following definitions. The idea is that I have a grid of squares, each of which is associated with a scalar value. Initially, only one square has a non-zero positive value, which can be any value but we might as well define it to be 1. You may think of this as a "drop of liquid". At every unit time thereafter, this "liquid" spreads to adjacent squares. Since there are 4 squares surrounding each square, the liquid spreads into 5 equal parts, one on the square it is on, and the others added to the neighbouring squares. We proceed this way, and after N iterations, we look at all squares with a value greater than H, for some positive H. such that H is less than the maximum value of all squares, and greater than some threshold (which I'm not sure how to define). The set of squares included will be a pixelated approximation to a circle---at least, this is what I think, I'm not sure how to prove it.

Anyway, enough hand-waving. Let's be more precise: let represent the value of the square (or n-cube) at coordinate at time . Let be the set of orthogonal unit basis vectors for . Let , where represents the zero vector. Then is defined recursively by:



Each value of represents the next iteration of the "spreading" of the "liquid" over the square (n-cubical) grid of . Let k\}" align="middle" />. The claim is that for large values of and certain values of ]]>
Anyway, I was experimenting with iterated algorithms and stumbled across the following observation, which led me to conjecture that perhaps it's generalizable to n-dimensions, and possibly to any kind of tiling of n-space. However, I'm not clear how to rigorously define my conjecture, so I'm posting here to see what people say.

First, let me explain the "intuitive" motivation behind the following definitions. The idea is that I have a grid of squares, each of which is associated with a scalar value. Initially, only one square has a non-zero positive value, which can be any value but we might as well define it to be 1. You may think of this as a "drop of liquid". At every unit time thereafter, this "liquid" spreads to adjacent squares. Since there are 4 squares surrounding each square, the liquid spreads into 5 equal parts, one on the square it is on, and the others added to the neighbouring squares. We proceed this way, and after N iterations, we look at all squares with a value greater than H, for some positive H. such that H is less than the maximum value of all squares, and greater than some threshold (which I'm not sure how to define). The set of squares included will be a pixelated approximation to a circle---at least, this is what I think, I'm not sure how to prove it.

Anyway, enough hand-waving. Let's be more precise: let represent the value of the square (or n-cube) at coordinate at time . Let be the set of orthogonal unit basis vectors for . Let , where represents the zero vector. Then is defined recursively by:



Each value of represents the next iteration of the "spreading" of the "liquid" over the square (n-cubical) grid of . Let k\}" align="middle" />. The claim is that for large values of and certain values of ]]> http://math.eretrandre.org/mybb/showthread.php?tid=17 Mon, 30 Apr 2007 22:26:37 +0200 http://math.eretrandre.org/mybb/showthread.php?tid=17
Putative proof:  Suppose the contrary.  Then let M(i, j) be the representation of such a well-ordering of all the OMEGA-random binary sequences in the open real interval, (0, 1), where  i,j  are indicial numerals, not necessarily belonging to any countable ordinal.  Consider the "Cantorian Diagonal", represented as M(k,k) where
k = i = j:  Assuming that M(k,k) is necessarily also a OMEGA-random binary sequence, the fact that the complement M'(k,k) differs from each element of M(i,k) in at least one component, hence, as defined, M'(k,k) is not included in M(i,j), and since the complement of any OMEGA-random binary sequence is another OMEGA-random binary sequence,  we have the sought-for contradiction which proves the theorem.  QED

Comment:  Since any consideration of the cardinal-magnitude of the index set was explicitly excluded in the statement of assumptions, it follows that the attribute of NON-well-orderability is LOGICALLY PRIOR to the attributes of countability/uncountability.

With that in mind, consider the contra-factual hypothesis under which the researches of Professors Gregory Chaitin, Per Martin-L\"ov, and Robert Solovay (establishing the existence and critical characteristics of OMEGA-random binary sequences) might have preceded Cantor's work: Then the (also contra-factual) hypothesis that the above given result might have also preceded Cantor's proof of the uncountability of the reals in (0, 1) would raise the (presently?) open question of whether there is any necessary correlation between the attributes of NON-well-orderability and NON-denumerability.

Quoting the last sentence of Henkin's doctoral dissertation, "But perhaps this is only Philosophy?"

Reference (1): G. J. Chaitin, "Algorithmic Information Theory", Cambridge University Press 1987.


Any comments?]]>
Putative proof:  Suppose the contrary.  Then let M(i, j) be the representation of such a well-ordering of all the OMEGA-random binary sequences in the open real interval, (0, 1), where  i,j  are indicial numerals, not necessarily belonging to any countable ordinal.  Consider the "Cantorian Diagonal", represented as M(k,k) where
k = i = j:  Assuming that M(k,k) is necessarily also a OMEGA-random binary sequence, the fact that the complement M'(k,k) differs from each element of M(i,k) in at least one component, hence, as defined, M'(k,k) is not included in M(i,j), and since the complement of any OMEGA-random binary sequence is another OMEGA-random binary sequence,  we have the sought-for contradiction which proves the theorem.  QED

Comment:  Since any consideration of the cardinal-magnitude of the index set was explicitly excluded in the statement of assumptions, it follows that the attribute of NON-well-orderability is LOGICALLY PRIOR to the attributes of countability/uncountability.

With that in mind, consider the contra-factual hypothesis under which the researches of Professors Gregory Chaitin, Per Martin-L\"ov, and Robert Solovay (establishing the existence and critical characteristics of OMEGA-random binary sequences) might have preceded Cantor's work: Then the (also contra-factual) hypothesis that the above given result might have also preceded Cantor's proof of the uncountability of the reals in (0, 1) would raise the (presently?) open question of whether there is any necessary correlation between the attributes of NON-well-orderability and NON-denumerability.

Quoting the last sentence of Henkin's doctoral dissertation, "But perhaps this is only Philosophy?"

Reference (1): G. J. Chaitin, "Algorithmic Information Theory", Cambridge University Press 1987.


Any comments?]]> http://math.eretrandre.org/mybb/showthread.php?tid=15 Thu, 22 Feb 2007 11:57:01 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=15 quickfur Wrote:But regarding polyhedral volumes... isn't it just a matter of dissecting the polyhedron into right pyramids and summing them up? This isn't too hard to do from the vertex coordinates.


But that dissection can not be really expressed in a formula (rather in an algorithm). To illustrate, consider the following.

Out of curiosity, I once computed the area of a triangle given by the length of its sides (you know it is determined by that data) and got the following (seemingly unknown) formula:



(Ok, I see thats not exactly what I asked before, but nevertheless the topic is general enough)

It seems for me that every simplex is exactly described by the set of its edge lengths. So how is the general formula for the n-simplex?]]> quickfur Wrote:But regarding polyhedral volumes... isn't it just a matter of dissecting the polyhedron into right pyramids and summing them up? This isn't too hard to do from the vertex coordinates.


But that dissection can not be really expressed in a formula (rather in an algorithm). To illustrate, consider the following.

Out of curiosity, I once computed the area of a triangle given by the length of its sides (you know it is determined by that data) and got the following (seemingly unknown) formula:



(Ok, I see thats not exactly what I asked before, but nevertheless the topic is general enough)

It seems for me that every simplex is exactly described by the set of its edge lengths. So how is the general formula for the n-simplex?]]> http://math.eretrandre.org/mybb/showthread.php?tid=14 Mon, 19 Feb 2007 03:49:53 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=14
Recently I tried to look up the formula for the circumference of an ellipse, and found that it is highly non-trivial (involving elliptical integrals). However, I wonder if the following special case may be easier to calculate: given a circle of fixed circumference C and radius r, how easy is it to find an ellipse E with one of the axes already given, such that E has circumference C? E.g., if I deform a circle into an ellipse with major radius R, what must the minor radius be in order for its circumference to be constant? Thanks.]]>
Recently I tried to look up the formula for the circumference of an ellipse, and found that it is highly non-trivial (involving elliptical integrals). However, I wonder if the following special case may be easier to calculate: given a circle of fixed circumference C and radius r, how easy is it to find an ellipse E with one of the axes already given, such that E has circumference C? E.g., if I deform a circle into an ellipse with major radius R, what must the minor radius be in order for its circumference to be constant? Thanks.]]> http://math.eretrandre.org/mybb/showthread.php?tid=13 Wed, 10 Jan 2007 12:25:47 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=13
I extremly enjoy the feature to include LaTeX code within "tex" / "/tex" tags. This produces inline formulas. Is there a possibility to get displayed formulas too (i.e. formulas shown on thier own lines with those lines numbered)?]]>
I extremly enjoy the feature to include LaTeX code within "tex" / "/tex" tags. This produces inline formulas. Is there a possibility to get displayed formulas too (i.e. formulas shown on thier own lines with those lines numbered)?]]> http://math.eretrandre.org/mybb/showthread.php?tid=12 Sat, 16 Dec 2006 21:47:29 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=12 http://math.eretrandre.org/mybb/showthread.php?tid=11 Sat, 16 Dec 2006 17:55:11 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=11
This is somewhat equivalent to that every increasing sequence (for the countable case) has a limit.

This post is due to time limits a bit a stub. But I see 3 possibilites to check


surreal/Conway numbers
hyperreals (non-standard analysis)
home made construction

]]>
This is somewhat equivalent to that every increasing sequence (for the countable case) has a limit.

This post is due to time limits a bit a stub. But I see 3 possibilites to check


surreal/Conway numbers
hyperreals (non-standard analysis)
home made construction

]]> http://math.eretrandre.org/mybb/showthread.php?tid=10 Sat, 16 Dec 2006 09:42:54 +0100 http://math.eretrandre.org/mybb/showthread.php?tid=10