Concerning:
bo198214 Wrote:GFR Wrote:The problem, as you also correctly said, is to define an operation (ONE hyperop) that would be the unique operation, exactly fitting in the hyperops hierarchy, at rank 0.
Only to summarize and clarify the current situation:
If we agree on the law a[n+1](b+1)=a[n](a[n+1]b) for all hyperoperations [n] for integer n and agree that a[1]b=a+b then it stringently follows (without assumptions about initial values) that a[0]b=b+1 and it also stringently follows for all hypo operations that a[-n]b=b+1. (This was shown in this thread by Andrew and me.) I would call this "exactly fitting in the hyper operations hierarchy".
Actually, I use to write the abovementioned "Mother Law" as:
a[n-1](a[n]b) = a[n](b+1), which gives:
a+(a*b) = a*(b+1)
a*(a^b) = a^(b+1)
a^(a#b) = a#(b+1)
and which, for n = 1, also gives:
a[0](a[1]b) = a[1](b+1), i.e.:
a ° (a+b) = a + (b+1) = a+b+1 = (a+b) + 1... so far, so good ... !
Let me try now to take a new (multiple...) way, starting from this initial conclusion, where, by putting a + b = k and reasoning, for the moment, only with positive integers, we should have:
a ° k = k + 1, with, obviously (but not compulsorily), k > a.
This multiple way (quadruple, not ... octuple) is of the
inductive and not of the deductive type. I hope that everybody would be patient enough to read it, without ... fainting, deciding to go to the Foreign Legion or (BO) organizing a metaphorical
Srafspedition for the democratic elimination of ... somebody from this Forum.
, I mean
Pillar 1 - The Mother Pillar. Supposing that the "Mother Law" means
exactly fitting zeration into the hyper operations hierarchy, then we could assume that:
a ° b = b + 1, apparently only depending on the second operand.
The problem here is that, in my humble opinion, this fact would demonstrate that a
zeration binary operation could not exist. Unless (there is always an ... unless) the Mother Law is not alone. I mean, it might be necessary, but not sufficient, or sufficient, but not necessary, or neither of them (but this would be too much!). In fact, it would be a nonsense just to say that a ° b is the successor of b, for any a. If we are looking for a new binary operation, we should be prepared to find other additional conditions, accompanying and supporting the Mother Law.
Pillar 2 - The Ackermann Pillar. We know that the Ackermann Function (AF) can be defined as follows:
A(0, n) = n+1
A(s, 0) = A(s-1, 1)
A(s, n) = A(s-1, A(s, n-1))
The AF can be shown as an infinite matrix, starting form line s=0 and column n=0, extended to all the natural n's and s's. Terrific landscape! Nevertheless, strangely enough and by using the hyperops formalism, any A(s, n) element of the AF matrix, for s>0, can also be shown as follows:
A(s, n) = 2[s](n+3) - 3
For instance:
A(1, 1) = 2 + 4 - 3 = 3
A(2, 1) = 2 * 4 - 3 = 5
A(3, 1) = 2 ^ 4 - 3 = 13
A(4, 1) = 2 # 4 - 3 = 65533
etc..., usw...
The first line of the matrix is a ... problem, because it is simply given by: A(0, n) = n + 1, while the general AF formula gives: A(0, n) = 2[0](n+3) - 3. In conclusion, for
n >= 0, we shold have:
A(0, n) = (2[0](n+3)) - 3 = n + 1, or (all bracketing is necessary), with
k = n + 3:
(2[0](n+3)) - 3 = n + 1 = (2[0]k) - 3 = n + 1, or:
2[0]k = n + 4 = k + 1, with n >= 0, i.e.: k > 2.
In conclusion, the Ackermann general formula A(s, n) = (2[s](n+3)) - 3 can be made valid also for line s=0 if we would define a
general rank zero operation of the type:
a[0]b = b + 1, if b > a, coinciding, for a = 2, with:
2[0]b = b + 1, for b > 2.
Under these conditions, we should have, for zeration:
a ° b = b + 1, if b > a
Now, the problem (... again!) is that zeration seems not to be defined in the case of b =< a, which, indeed ... and again, is not acceptable. We need more pillars.
Pillar 3 - The Hyper-roots. It is known, from the Ancient Greeks' times, that the square root of a number can be calculated by iterating the following functional equation:
y = sqrt x ---> (y + x/y) / 2 => y
Iteration (n + x/n) / 2 = m -> n, starting from an approximate solution n, rapidly converges to the square root of x. About 20 years ago, Konstantin Rubtsov, thought to apply a similar formulation for calculating the square superroot, as well as the half of a number (!!), both left-inverse hyperops, of the root type. The compact formulation of that can be generalized as follows:
y = x /[s]2 ---> y <= (y[s-1](y[s]\ x)) /[s-1]2.
This formula can be implementes as follows:
.....
y = ssqrt x ---> y <= sqrt (y * log_y(x))
y = sqrt x ----> y <= (y + x/y) / 2
y = x / 2 -----> y <= (y ° (x-y)) - 2
It can be easily verified that the square superroot (super square root) and the square root are rapidly converging to an acceptable value after few iterations. Concerning the formula including zeration, the situation is that:
- "y" must be an even natural number, for allowing us to find its half;
- zeration must be commutative, for allowing us to calculate the approximate values, for any initial "y".
The first condition is due to the fact that zeration has been initially defined only for integer numbers and that the formula needs an even number for calculatind its half. The second condition has a deeper meaning, because it just
doesn't converge if zeration is not commutative. The conclusion is that the second condition is fulfilled only if the order of the operand can be commuted, i.e. if:
a ° b = b ° a = max(a, b) + 1, if a >< b.
Now, commutativity of zeration is one of its most important properties, if fully and surely demonstrated. Unfortunately, the abovementioned "speach" is not a rigorous demonstration, but a cloudy (quick and ... dirty
) mathematical experiment. Konstantin Rubtsov (Rubcov) knows a complicated, but very "clean" demonstration of the commutativity of zeration, based on the consideration of the left and right neutral elements, homomorphism with addition and/or multiplication, cathegory theory and ... other similar amenities. It takes several DIN A4 pages, like the Goedel's Theorem, and any shorter presentation is just hermetical. We should convince him, when he shall have time, to present a "people's demokratic" version of it, for simple minded guys, like me. For the moment, I keep the Faith, thinking that, after all, we could consider that statement as part of a postulated axiom (BO ipse, in a moment of ... weakness, dixit!).
No instructions are given if a = b. However, this condition doesn't contradict those of the other pillars, but it completes them, despite the fact that the constraints under which it has been discussed are a little bit weak. This pillar is, nevertheless, reiforced by the following one.
Pillar 4 - The Hyper-means. Standard Algebra has defined, since a long time important binary operations such as the arithmetic and the geometric means, strongly associated with two important classical hyperops, i.e.: addition and multiplication, as follows:
am(a, b) = (a + b) / 2, with: am(a, a) = a (the arithmetic mean)
gm(a, b) = sqrt(a * b), with: gm(a, a) = a (the geometric mean).
In the hyperops hierarchy framework, we can also coherently define two other hyper-means, the power and the zeric mean, such as:
pm(a, b) = ssqrt(a ^ b), with pm(a, a) = a (the power mean)
zm(a, b) = (a ° b) - 2, with zm(a, a) = a (the zeric mean).
The problems with the power mean is that it operates on a non commutable operation (exponentiation), on one hand, so that pm(a, b) >< pm(b, a). On the other hand, ssqrt(x) has real values only for x > e^(-1/e). For these reasons, despite its importance for studying possible fractional hyperop ranks, its analysis requires further attention. On the contrary, the zeric mean appears also in Pillar 3 and this strongly justifies the fact that it must be:
a ° a = a + 2
This provisional and partial conclusion also justifies the following (almost initial) relations, which were at the origin of each research work made in the hyperops field:
.........
a ^ a = a # 2
a * a = a ^ 2
a + a = a * 2
a ° a = a + 2
........
as well as, more particularly (for a = 2), the almost "holy" tetragonal equality:
.... 2 ° 2 = 2 + 2 = 2 * 2 = 2 ^ 2 = 2 # 2 = ..... 2 [s] 2 .... = 4 !!!!!!
Conclusions based on the Four Pillars - Based on the Four Pillars, the existence of a zeration hyperop could be justified by the following definitions (considered as a postulate by BO and as a consequence of the overall existing mathematical environment, by KAR):
a ° b = max(a, b) if a >< b
a ° b = a + 2 = b + 2 if a = b
Personally, I think that this definition just satisfies the constraints put forward by the four Pillars and, for this reason, it justifies the KAR's zeration. I have some doubts concerning its unicity, but this happens in the best families, such as the Euler's Gamma function as extension of the factorial.
For my intellectual equilibrium, I use to pronounce "over" (or "more" or "beyond") the "°" zeration infixed operator, like "plus" or "(multiplied) by" in case of addition and multiplication. This gives, expressing it as a prefixed operator (just for ... confusing a little bit the reader):
over(a, b) = max(a, b) + 1, if a >< b
over(a, b) = c + 2, if a = b = c.
Konstantin doesn't like that, because he thinks that this would create an additional confusion with operator "+", but I feel very cool when I pronouce it so. Under these conditions, the zeric mean would be:
zm(a, b) = max(a, b) - 1, if a >< b
zm(a, b) = c, if a = b = c.
The Neutral Elements - What somebody calls the left/right unit elements. Let us consider the following general functional equation, including the definition of a fixpoint of the type x = f(x):
x = a[s]x <---> x = a[s+1]oo, which means:
a = x /[s]x <---> a = x /[s+1]oo (inverses, of the root types)
The implementation of that for ranks 3, 2, 1, 0 gives:
x = a^x <---> x = a#oo
a = x-rt x <---> a = oo-srt x = x^(1/x) (no neutral element for rank 3)
x = a*x <---> x = a^oo
a = x/x = 1 <---> a = oo-rt x = x^0 = 1
x = a+x <---> x = a*oo
a = x-x = 0 <---> a = x/oo = 0
x = a°x <---> x = a+oo
a = xçx <---> a = x-oo = -oo (ç stands here for the delta symbol, inverse of "°")
From the hyperops point of view, such fixpoint can be used (if it is defined via a constant) for the definition of the left neutral element (of the root type). In fact, we have:
-- x = a^x: no left neutral element for exponentiation (a = x^(1/x));
-- x = a*x means a = 1;
-- x = a+x means a = 0;
-- x = a°x means a = -oo.
The -oo element is, therefore, the left neutral element of zeration and, due to its commutativity, also the right one and we have:
(-00)°a = a°(-oo) = a.
Final Remarks- Please find in annex two plots of y = 2°x and of its inverse y = 2çx = 2-delta-x, according to the abovementioned KAR definitions. Please note:
(1)- Zeration y = a ° n, with n natural, is defined for -oo =< n =< +oo and it can be considered as a (one-valued) "function" .
(2)- Zeration can easily be extended to the real numbers but it is absolutely not a contimuous function and, therefore, it is not analytic (as somebody initially suggested); in particular, it contains a single "spot" (one separate point) function value and an "infinite discontinuity", both characterizing its clear and deep discontinuity. This fact should not impress people knowing "functions" like the Dirac and the Step and the Ramp functions, as well as other strange discontinuous mathematical objects.
(3)- Zeration y = a ° x is inversible (Deltation) but this fact opens the "door" to new classes of numbers, such as the non-standard trans-finite numbers and also the rather new hypothetical "
trans infinite numbers" (
Delta Numbers, according to KAR's terminology). This fact should not impress people knowing all the secrets of the logarithms of negative and/or complex numbers (or of logarithm "
tout-court"), which involve the "wild" complex multi-valued "numbers".
(4)- The down-up methodology used in this thread implies a sort of "experimental mathematics", which should not impress distinguished Researchers familiar with these procedures. Take, for instance the ghost hunting programs lauched to find complex (i.e.: unreal and almost ... non existing!) fixpoints.
(5)- The KAR's definition (postulate or experimental finding) were published in 1987 and 1996 and presented in two World Congresses of Mathematicians (held in Zurich and Madrid .... I don't remember the dates), as well as in the WRI Forum and no criticism was received concerning that, ... until now.
(6)- We should avoid to state that an operation with multiple solutions
is not defined. Think, for instance, of the number of solution of the square root of 4, of the cubic root of 8 or of the fourth root of 16, which are two, three and four, respectively. "Disequation" such as: real x > 5 defines all the real numbers greater than 5. If we think of that formula as an "operation", the number of its solutions is a non-coutable infinity. Try also, for instance, to draw the "plot" of y = x ^ e. Please don't tell me that these things are not in the ... "Manual".
(7)- This is the moment of re-analyzing the bases of those Pillars and definitions, in the framework of the perspective developments in the field of tetration and of the overall hyperops hierarchy. Nevertheless, the analysis should be carefully and precisely done, trying to avoid easy simple criticisms to the "strangeness" of the entire ... business. It would be interesting to find a new "pattern", functioning as additional pillar for the entire hierarchy. Why not ... !
Please consider these notes as a personal comment, concerning only me and not KAR, who is very busy in this moments of his professional life. Thanks to all of you for your kind attention and for your useful cooperation in this enterprise.
I stop here, apologizing for my macaronical English and for all the repetitions and possible errors. I also do it for giving to the Administrator space and time to his usual and friendly "But Gianfranco ..." routine.
!!
Gianfranco
, 118-133]