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Stability of I in tetration and scaled infinities/infinitesimals?
#1
I will try to ask this question more correctly as it bothers me:

If we apply Mother Law of hyperoperations:

,

to infinite application of operations with than:





Tetration is the first operation where something qualitatively new happens- I is returned as a result of infinite application of operation:



as

This allows(?) to apply formula recursively, so that by denoting :



If this is continued infinitely, then in the end ( I have elaborated it here :Application of infinite hyperoperations to I^(1/I) .



I wonder is this true as it got quite complex in the end.

If so, is very strong attractor for infinite tetration of , as any type of infinity via operations above tetration returns I.

How I understand it, if we reverse the operations, any smallest, infinitesimal deviation from will return a result infinitely away from .

If we had looked only at :



we would not noticed how extremely, sensitive is infinite tetration to deviations from or as input and output.

I my opinion, that allows to consider or define infinitesimal deviations from and however strange it may sound and look.

The basic property of such infinitesimal deviation would be divergence from and if all operations are applied.

Howerever, as seen from mother law, if is applied to the tetration of than it will still converge to I if the deviation from is the smallest possible special infinitesimal , however, it will diverge if the deviation is still infinitesimal ( as somehting infinitely small) , but any other (bigger than?) EXCEPT the smallest one.

Excuse me for strange ideas, but I believe both infinitesimals and infinities are scaled and has structure, and very exactly so.

I would love to hear if such ideas connect to some existing mathematical constructs.

Tetration seems the right operation, fast enough to start to discuss such ideas as structures and scales of infinities/infinitesimals.

Of course, it seems one can use any number of selfroot type in these formulas, and that is true, except that and seems to be the only basic combination involving purely imaginary number and purely real number .

Ivars
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#2
One more observation about constants as such:

If we apply infinite (with a> e^(1/e) and even finite (when a<1) real tetration to some numbers and get complex numbers as a result, the constancy of such number a vs. tetration becomes very questionable.

Numbers like e , pi , i , irrationals, periodic decimals etc are not really constants but symbols, as they are incomputable, like infinity and 0.


So far, they have behaved like constants under differentation/integration as long as 3 usual operations only have been involved.

There is no reason to think they remain constants (i.e. has no internal structure) if tetration and higgher operations,as well as intermediate (fractional) and zeration is included.

If there are reasons, I would be happy to hear about them.

Ivars
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#3
Sorry Ivars but it just seems as this is a restart of the discussion about different infinities/infinitesimals. If you so believe in it, why not take a lesson in non-standard analysis. However for the classical approach to analysis there is only one infinity in:



and in that context it makes no sense to continue a useful discussion.
If you were inspired by quickfur's posts then thats again another thing, ordinal numbers. Here however again the advice: become familiar with (at least the basis of) the topic before starting with wild speculations.


Ivars Wrote:Numbers like e , pi , i , irrationals, periodic decimals etc are not really constants but symbols, as they are incomputable, like infinity and 0.

Infinity is not a number, 0 is a number. e, pi, i, irrationals and periodic decimals are numbers too. There are uncomputable irrational numbers, however e, pi, i and periodic decimals are computable (in the sense that there is an algorithm to determine the n-th digit after the point).

Quote:So far, they have behaved like constants under differentation/integration as long as 3 usual operations only have been involved.

There is no reason to think they remain constants (i.e. has no internal structure) if tetration and higgher operations,as well as intermediate (fractional) and zeration is included.

This is mystics and mathematical nonsense with no base.
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#4
OK. I will make next attempt to peek into infinitesimals/infinity/i/e/pi/0 after improving my qualification.

Ivars
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#5
Perhaps these operation sequences or similar can be used to define different infinities since each of them has own speed of growth.

Ivars
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#6
bo198214 Wrote:
Ivars Wrote:Numbers like e , pi , i , irrationals, periodic decimals etc are not really constants but symbols, as they are incomputable, like infinity and 0.
Infinity is not a number, 0 is a number. e, pi, i, irrationals and periodic decimals are numbers too. There are uncomputable irrational numbers, however e, pi, i and periodic decimals are computable (in the sense that there is an algorithm to determine the n-th digit after the point).
Dear Ivars,

Unfortunately (ops ... , sorry Bo Wink , I didn' mean that!) I think that Henryk is absolutely right. The entities that you mentioned are indeed constant. Where did you find the idea that they are only "symbols"? They are constant numbers, ... because they are not variable. We should not confuse "variability" with the technical impossibility of "fully knowing them". If a number is computable, that means that it is mathematically known, i.e. that "there is an algorithm to determine the n-th digit before and after the point".

The fact that, techically, we don't have an infinite computer memory for storing ... all ... its representative figures (say, by using the decimal fixed point notation) is an engineering and not a mathematical problem. Engineers say that Pi cannot be fully known (or only by a certain approximation), but mathematicians say that it CAN be known, if we take the appropriate (infinite) number of necessary steps. For a math-man, it is sufficient to demonstrate that the algorithm exists. Then, the number is known. There are also cases where we can demonstrate the existence of a number, but we cannot compute it. But, in all these cases, it is out of question to consider them as variables or, ... just symbols.

Believe me! The decimal figures of Pi are not random, they are deterministically determined, if you see what I mean. The fact that an American Corporation provided in the past "pseudo-random" sequences of figures based on the decimal figures of PI, for random testiong nuclear reactor cores (Monte Carlo method), is another story. We are only human beings and we don't know what we are doing. For the non-destructive testing it was sufficient, but the sequence was not random.

From another point of view, expression e^(i*Pi/2) = i is a well known equality (identity) and it does not mean at all that i has .. an internal structure. Actually, i = sqrt(-1), with plus/minus, if you wish.

Please finally accept the fact that |d(Pi)| = 0. Don't try to differentiate constants, you will loose time ... for nothing. Move higher, as I already friendly suggested you, ... there is a lot of space!.

GFR
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