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Laws and Orders
#11
andydude Wrote:I recognize that both the GML and ML as fundamental properties of hyperoperations. I also recognize that, although less intuitive, the Balanced Mother Law (BML) is very successful at producing the usual hyper1, hyper2, hyper3 as we know them. If you are responding to my comment in that thread, then I was only talking about BML, not the other two. I think so far we have established that the GML-hyper-0 is "zeration" as you define it, ML-hyper-0 is "succession" and BML-hyper-0 is the empty set.
I agree. So, for the moment, I think we have the following "laws", sometime incompatible among them, particularly "near" rank "0" (I use s and r, just to see that I have correctly understood):

GML - the Grandmother Law: ...... a[s]<r>a = a[s+1](r+1)
ML.. - the Mother Law: ............. a[s](x+1) = a[s-1](a[s]x)
BML. - the Balanced Mother Law : a[s+1](2x) = (a[s+1]x) [s] (a[s+1]x)
DL ..- the Daughter Law: ........... a[s]x = x = a[s+1]oo

Within this list, I'd like to draw the general attention to a relation "naturally" derived from GML, for r=1 (by a simple mono-iteration of [s]): a[s]a = a[s+1]2, automatically valid when GML holds.

The beautiful thing about that is that, for a=2, we get: 2[s]2 = 2[s+1]2 !!!! . Now, since we know for sure that:
2[4]2 = 2[3]2 = 2[2]2 = 2[1]2 = 4, i.e.:
2#2 = 2^2 = 2*2 = 2+2 = 4
then, we can quietly (!) assume that we should always have:

NL .. - 2[s]2 = 4 , for any (... integer?) rank s. I'd like to call this the "Niece Law", because is very close to the GML.

@Ivars: In other words, we should also have (if these operations ... exist):
2[-1]2 = 4 (minusation) !! and
2[w]2 = 4 (omegation) where w (omega .. !) is the first infinite, coutable, ordinal number.

@Andydude & Jaydfox: Please add 2[w]2 = 4 to your analysis of e[w]n, with negative n, apparently giving:
e[w]n = n-1. ..... That's ... really all, Folks ... ????? ..... Much ado about nothing.

The hypothetical cases of 2[i]2 = 4, as well as of 2[0.5]2 = 2[1.5] = 2[2.5]2 = 4, should be supported by more serious considerations. However, ... why not??!! We shall see. Smile

It would be interesting to study the feasibility of "half-way rank" hierarchy [s=(2n+1)/2, with n relative integer], with the only link to the "mathematical reality" given by the "Gauss Mean", at rank s=3/2.... . Definitely so.

GFR
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#12
GFR Wrote:@Ivars: In other words, we should also have (if these operations ... exist):
2[-1]2 = 4 (minusation) !! and
2[w]2 = 4 (omegation) where w (omega .. !) is the first infinite, coutable, ordinal number.

The hypothetical cases of 2[ i]2 = 4, as well as of 2[0.5]2 = 2[1.5] = 2[2.5]2 = 4, should be supported by more serious considerations. However, ... why not??!! We shall see. Smile

GFR

Do I really have to read about those ordinals? They sound scary.With what purpose? I think numbers have fine structure, is it explained by these ordinals?

What about i [ i] i ? Value and meaning? Related to :

0[0]0
1[1]1
2[2]2
3[3]3
4[4]4

x[x]x

i[ i]i

z[z]z

q[q]q where q is quaternion...

I think I know what is "imagination" . It is , even if I know you will say it can not be , related to ordering/enumeration in time. As opposed and complementing to ordering in space, or dimension(s) represented by real hyperoperations.

I have to think what then it means when applied to various bases and simple cases.

Ivars
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#13
GFR Wrote:GML - The Grand-mother Law, considered by all people approaching this matter for the first time, as a first definition of the hierarchy, after applying the "priority to the right" traffic rules to the hyperops operators. It sounds like:
a[s]<r>a = a[s+1](r+1).

Unfortunately the naming is against its context!
What you call grand mother law is rather the daughter law of the mother law, in the sense that it is a particular case of the mother law for those operations with the initial condition a[s+1]1=a:

From GML directly follows:
a=a[s]<0>a=a[s+1]1

Actually you cant use it to define a zeration because for the above equation is wrong/not applicable. Yes, is wrong. The context here is what you call an equality. You have an equality (your GML) that means that the equation is true for all assignments of the contained variables from their corresponding domain of definition. Then we put the particular value and into the GML and gain the *equality* , i.e. it must be true for all of the domain of definition (which I assume is the real numbers). But instead it is not even true for one ! So the GML is not true for .

Quote:DL - The Daughter Law (this is a ... new one !! Haha! Well, not really!). In fact, the recursive application of the hyperops operator gives, as a consequence of situation "b":
a[s]x = x = a[s+1]oo.

The daugher law should correctly be stated as:

If a[s+1]oo=x then a[s]x=x.

As I already showed in the Zeration thread what you call DL is a consequence of your GML and
the more general DL that follows from the mother law (i.e. is valid for *all* operations not only those with x[s+1]1=x) is

If a[s+1]oo=x then a[s]x=x[s+1]1.
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#14
bo198214 Wrote:From GML directly follows:
a=a[s]<0>a=a[s+1]1

Actually you cant use it to define a zeration because for the above equation is wrong/not applicable. Yes, is wrong. The context here is what you call an equality. You have an equality (your GML) that means that the equation is true for all assignments of the contained variables from their corresponding domain of definition. Then we put the particular value and into the GML and gain the *equality* , i.e. it must be true for all of the domain of definition (which I assume is the real numbers). But instead it is not even true for one ! So the GML is not true for .
Thank you for your comments, Henryk!

What I am trying to say is that:
a°a = a+2, for GML
a°a = a+1 for ML
a°a = a[0]<0>a, for GML and ML supposed together valid.
and that, therefore:
a[s]<0>a = a , for any s>0 is a right equality, but:
a[0]<0>a = a is a wrong equality.

So, your point of view is that, supposing always valid ML, GML (that you derived from ML) collapses for s=0 and, therefore, it cannot be taken as the definition of zeration. My point of view is that, if you take GML as initial "source law" and as descriptive statement for the definition of the hierarchy (in this case, ML would be one of its "properties"), then we have to accept the fact that ML may collapse for s=0 and, therefore that a°a = a+1 is a wrong equality, violating the initial general definition of the hierarchy.

I think that also Andydude agrees with that. He said that the choice between GML and ML as definition of the hierarchy is indeed a matter of choice (or something like that). Actually, we should have two versions of the "level zero" operation and what I call "zeration" is the version considering GML as the definition of the hierarchy.

In other words, I (& al.) started asking to ourselves:
"Do we have hyperoperation hierarchy definable by: a[s]<r>a = a[s+1](r+1) ?"
This means that I (& al.) started by tye GML. For r=1, we have:
........
a[4]a = a[5]2 ----> a#a = a$2 (hypothetical pentation)
a[3]a = a[4]2 ----> a^a = a#2 (our tetration research)
a[2]a = a[3]2 ----> a*a = a^2 (OK, known)
a[1]a = a[2]2 ----> a+a = a*2 (- ditto -)
a[0]a = a[1]2 ----> a°a = a+2 (this is what we were looking for)
a[-1]a = a[0]2 ---> .............. (very strange, I must admit).

Now, the answer to that could be: YES, we have such a hierarcchy, or: NO, we have not such a hierarchy, but we may have something else. If the answer is that we have two or more versions of it, ... I shall feel really in trouble. We should try with a half-rank operation hierarchy and see which is the stronger initial statement (GML or ML), allowing the "chain" to survive. I mean which is the law allowing that half-way hypothetical hierarchy to exist. I discussed with Ingolf Dahl (participant in this forum) about that, some time ago. By the way, how are you, Ingolf?

I know what I said concerning the Ackermann Function (AF). But, Henryk, shall we consider that pillar as the most fundamental of the hyperops hierarchy, or just a proof of its usefulness for the description of the AF elements. In this case, we shall not be surprise that ML faints for s=0.

Just a hypothesis. What do you think?

GFR
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#15
Concerning the DL, I prefer my "implication", i.e.:
IF a[s]x = x, THEN x = a[s+1]oo
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#16
GFR Wrote:In other words, I (& al.) started asking to ourselves:
"Do we have hyperoperation hierarchy definable by: a[s]<r>a = a[s+1](r+1) ?"

And the answer is "No!" as I already showed (if we allow s=0). We are a group of mathematicians, it does not matter if someone has this opinion or that opinion if the facts are different. So again:

We start with the law
a[s]<r>a = a[s+1](r+1)

then I put r=0, as a[s]<r> is the r times iteration of f(x)=a[s]x, it follows that a[s]<0> is the identity, a[s]<0>x=x, so:

a = a[s]<0>a = a[s+1]1

now we set s=0:

a = a[1]1

as we agree that in our operation sequence x[1]y=x+y, the above is a contradiction.
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#17
Dear Henryk,

Ok! I just doubted that we can put s=0 in a[s]<0>, without warning that perhaps (exceptionally), and in this particular case, a[0]<0> is not the identity operator. If this is the case, the fact that you mentioned would not exist. Like the situation we find by ... dividing something by 0, or by trying to calculate a^0, for a=0. For s=0 something is collapsing. My curiosity is to see what exactly does (I mean ....: collapse).

Well, you are the Organizer of this Forum and its Moderator and you are right in trying to moderate me. I appreciate that. But your adamant (and certainly justified) stipulations recall me a funny story. Two men are writing on a door: "If you touch the (electric) wires you will die ... and go to Hell!". Then, they suddenly stop and one says:... "Perhaps, we have exagerated a little bit. It was sufficient to say that he will die".

Now, please, you may keep aside all unappropriate weapons. I shall keep quiet and cool for a while, about the zeration business.

GFR
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#18
GFR Wrote:Concerning the DL, I prefer my "implication", i.e.:
IF a[s]x = x, THEN x = a[s+1]oo

Gianfranco, that is not true!
Take and then surely . But . You can not bend mathematics to suit your wishes.
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#19
GFR Wrote:I just doubted that we can put s=0 in a[s]<0>, without warning that perhaps (exceptionally), and in this particular case, a[0]<0> is not the identity operator.

However I never saw a case where . What is a[s]<0> then in your opinion?

Quote:Well, you are the Organizer of this Forum and its Moderator and you are right in trying to moderate me.
...
Now, please, you may keep aside all unappropriate weapons.
...

Hey, hey, I never moderated you in any way, nor did I mention the possibility. You could and can write freely without restriction what you want on this forum (of course obeying the basic netiquette rules).

I merely want to convey that establishing mathematical theories is not like convincing your sponsor to give money. Certain rules that work well in social communication - for example the more often and by the more people a certain statement is repeated, the more true it becomes - dont work in mathematics. And also building lobbies and parties to support or enforce the own opinions does not work. Quoting authorities does not work either.

Basically it is about providing/proving theorems about definitions which you are free to design yourself. Assessing the beauty and propagating is then again subject to opinions and social communication.
But the start has to be true theorems/derivations.

So the easiest way for you would be just to exclude the case in your GML, however whether this increases the beauty is another question. But then zeration is still not determined by the GML (as I showed in the zeration thread, and there are still no uniquness conditions for *your* zeration provided). So what Andrew said in a side sentence:

Quote:I think so far we have established that the GML-hyper-0 is "zeration" as you define it

is not true, as there is no the zeration following from the GML nor is there the zeration following from GML plus additional conditions. While there is the zeration following from the ML.

Quote:I shall keep quiet and cool for a while, about the zeration business.

That is absolutely not necessary.
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#20
OK! Formally, I was, ... again ...., again joking. Substantially, I agree with what you said about mathematical truth. I just repeated some things because I was convinced of them and thought not to be clear enough and not at all for installing a kind of "Propaganda Abteilung" routine. I shall keep a little bit quiet for avoiding unnecessary and useless spamming. However, I have always doubts, about ... everything. Sorry about that. "Ich bin der Geist der stehts verneint".

GFR
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