Le 26-mai-06, à 19:35, Tom Caylor a écrit :
>
> Bruno Marchal wrote:
>> Hi,
>>
>> OK, let us try to name the biggest natural (finite) number we can, and
>> let us do that transfinite ascension on the growing functions from N
>> to
>> N.
>>
>> We have already build some well defined sequence of description (code)
>> of growing functions.
>>
>> Let us choose the Hall Finney sequence to begin with (but the one by
>> Tom Caylor can be use instead).
>>
>> F1 F2 F3 F4 F5 ...
>>
>> With F1(n) = factorial(n), F2(n) = factorial(factorial n), etc.
>>
>> Note this: Hal gave us a trick for getting from a growing function f,
>> a
>> new function growing faster, actually the iteration of the function.
>> That is, Hal gave us a notion of successor for the growing function.
>> Now the diagonalization of the sequence F1 F2 F3 F4 ..., which is
>> given
>> by the new growing function defined by
>>
>> G(n) = Fn(n) + 1
>>
>> gives us a growing function which grows faster than any Fi from Hal's
>> initial sequence. Precisely, G will grow faster than any Fi on *almost
>> all* number (it could be that some Fi will grow faster than G on some
>> initial part of N, but for some finite value (which one?) G will keep
>> growing faster. Technically we must remember to apply our growing
>> function on "sufficiently big input' if we want to benefit of the
>> growing phenomenon. We will make a rough evaluation on that input
>> later, but let us not being distract by technical point like that.
>> The diagonalization gives an effective way to take the "limit" of the
>> sequence F1, F2, F3, ...
>>
>> G grows faster than any Fi. Mathematician will say that the order type
>> of g, in our our new sequence F1 F2 F3 ... G, is omega (the greek
>> letter).
>>
>
> Bruno,
> You are starting to perturb me! I guess that comes with the territory
> where you're leading us.
You should not worry too much. I confess I am putting your mind in the
state of mathematicians before the Babbage Post Markov Turing Church
discovery. Everything here will be transparently clear.
> But of course being perturbed doesn't
> necessarily imply being correct. I will summarize my perturbation
> below. But for now, specifically, you're bringing in transfinite
> cardinals/ordinals.
Only transfinite ordinal which are all countable, and even nameable,
for example by name of growing computable functions as I am
illustrating.
Be sure you understand why G is a well defined computable growing
function, and why it grows faster than each initial Fi. If you know a
computer programming language, write the program!
> This is where things get perverse and perhaps
> inconsistent. For instance, couldn't I argue that G is also infinite?
In which sense? All functions are infinite mathematical object.
Factorial is defined by its infiinite set of inputs outputs: {(0,1)
(1,1)(2,2) (3,6) (4,24) (5,120) ...}.
> Take n = some fixed N>1. Then F1(N) > 1, F2(N) > 2, F3(N) > 3, ...
> and Fn(N) > n, for all n. So each member of the whole sequence F1, F2,
> F3 ... G is greater than the corresponding member of the sequence 1, 2,
> 3, ... aleph_0 (countable infinity). Thus, G >(=) countable infinity,
> even for a fixed n=N>1.
You are right but G is a function. Actually it just does what it has
been programmed to. I don't see any problem here.
>
>> But G is just a well defined computable growing function and we can
>> use
>> Hall Finney "successor" again to get the still faster function, namely
>> G(G(n)).
>>
>> The order type of G(G(n)) is, well, the successor of omega: omega+1
>>
>> And, as Hall initially, we can build the new sequance of growing
>> functions (all of which grows more than the preceding sequence):
>>
>> G(n) G(G(n)) G(G(G(n))) G(G(G(G(n)))) etc.
>>
>> which are of order type omega, omega+1, omega+2, omega+3, omega+4,
>> etc.
>>
>> Now we have obtained a new well defined infinite sequence of growing
>> function, and, writing it as:
>>
>> G1, G2, G3, G4, G5, G6, ... or better, as
>>
>> F_omega, F_omega+1, F_omega+2, F_omega+3
>>
>> just showing such a sequence can be generated so that we can again
>> diagonalise it, getting
>>
>> H(n) = Gn(n) + 1, or better
>>
>> H(n) = F_omega+n (n) + 1
>>
>>
>> Getting a function of order type omega+omega: we can write H =
>> F_omega+omega
>>
>> And of course, we can apply Hall's successor again, getting
>> F_omega+omega+1
>> which is just H(H(n), and so we get a new sequence:
>>
>> F_omega+omega+1, F_omega+omega+2, F_omega+omega+3, ...
>>
>> Which can be diagonalise again, so we get
>>
>> F_omega+omega+omega,
>>
>> and then by Hal again, and again ...:
>>
>> F_omega+omega+omega+1, F_omega+omega+omega+2, F_omega+omega+omega+3
>>
>> ...
>>
>> Oh Oh! a new pattern emerges, a new type of sequence of well defined
>> growing functions appears:
>>
>> F_omega, F_omega+omega, F_omega+omega+omega,
>> F_omega+omega+omega+omega,
>>
>> And we can generated it computationnaly, so we can diagonalise again
>> to
>> get:
>>
>> F_omega * times omega,
>>
>> and of course we can apply Hal's successor (or caylor one of course)
>> again, and again
>>
>> ....
>>
>> Oh Oh Oh Oh Oh .... A new pattern emerge (the Ackerman Caylor one, at
>> a
>> higher level).
>>
>> F_omega,
>> F_omega + omega
>> F_omega * omega
>> F_omega ^ omega
>> F_omega [4] omega (omega tetrated to omega, actually this ordinal got
>> famous and is named Epsilon Zéro, will say some words on it later)
>>
>> F_omega [5] omega
>> F_omega [6] omega
>> F_omega [7] omega
>> F_omega [8] omega
>> F_omega [9] omega
>> F_omega [10] omega
>> F_omega [11] omega
>>
>> ...
>>
>> In this case they are all obtained by successive diagonalzations, but
>> nothing prevent us to diagonalise on it again to get
>>
>> F_omega [omega] omega
>>
>> OK, I think the following finite number is big enough:
>>
>> F_omega [omega] omega (F_omega [omega] omega (9 [9] 9))
>>
>>
>> Next, we will meet a less constructivist fairy, and take some new kind
>> of "big leap".
>>
>> Be sure to be convinced that, despite the transfinite character of the
>> F_"alpha" sequence, we did really defined at all steps precise
>> computable growing functions ... (if not: ask question please).
>>
>
> It seems to me that you are on very shaky ground if you are citing
> transfinite numbers in your journey to showing us your ultimate
> argument.
Please Tom, I did stay in the realm of the finitary. Even intutionist
can accept and prove correct the way I named what are just big finite
number. I have not until now transgressed the constructive field, I
have not begin to use Platonism! There is nothing controversial here,
even finitist mathematician can accept this. (Not ultra-finitist,
though, but those reject already 10^100)
> I also think that if you could keep your arguments totally
> in the finite arena it would less risky.
I have. You must (re)analyse the construction carefully and realize I
have not go out of the finite arena. Ordinals are just been used as a
way to put order on the successive effective diagonalizations. Those
are defined on perfectly well defined and generable sequence of well
defined functions. I have really just written a program (a little bit
sketchily, but you should be able to add the details once you should a
programming language).
> For instance, I think it is
> very risky to use things like the axiom of choice as part of a theory
> of everything.
Tom, I am trying to be 100% serious. I am giving you the "real thing".
If you think I am using the axiom of choice, you better tell me where.
I don't.
> If you have Arithmetical Platonism as one of your
> hypotheses, then you need to stick to parts of math that are generally
> agreed to be a part of the Platonic world and not a man-made tool.
I was not doing any piece of philosophy. Just a part common in science
and enginering: writing a program which, if ever it would be able to
run, would stop on a (very big) finite number. It is a problem in
computer science, and I have illustrated a "modern" solution, if you
want.
It is indeed the effectivity (mechanizability, programmability, ...) of
the diagonalization which makes possible to write programs encoding
tranfinite ordinals, and here I show how to use that fact to solve a
"concrete" problem: naming a *very* big number in a program having some
reasonable length.
> Of
> course you'll counter that even man-made tools, in fact anything that
> we can think of (that's consistent), is part of reality (assuming the
> multiverse). But the problem is consistency. In the realm of the
> infinite your playing with truth that is not necessarily provable, or
> part of the universe.
I insist, I did not leave the realm of the finite era, at all. There is
nowadays nothing philosophical here. Usual Analysis textbook used far
more less finitary notion.
Please try to circumscribe more clearly the point where you feel we
have leave the finite era, because if you will miss what will happen
later were indeed we will leave the finite or effective era. At some
point, we will be obliged to leave the finite era, and it will be a key
step, but until now we haven't: we have just written a program naming a
very big number.
>
> So you've got me stuck right here. I think that at this point, you're
> assuming your conclusion: everything is number.
But we are not even thinking on things like that. People could even
think I am leading to the contrary conclusion. Look the tricky problem
below:
>
>> Tricky Problem: is there a sequence in which all growing computable
>> functions belong? Is it possible to dovetail on all computable growing
>> functions, ...
Diagonalization seems to illustrate the impossibility of sequencing
effectively all computable growing functions: having a sequence like F1
F2 F3 F4 ..., we can find by diagonalization a new function not in the
list. But dovetailing seems to necessitate such a sequence, so it seems
we can't dovetail on all growing functions, but then how could a
universal dovetailer exist?
This is an exercise! Not a philosophical question. It is a crucial
exercise the solution of which will eventually justify the
differentiation between the logic G and G*.
Before tackling this problem, which has been tricky at a time of course
(but is transparently clear today for the experts) please be sure we
have not yet leave the finite and effective era. Don't worry I will
present other diagonalizations. It is really the "heart of the matter".
Bruno
http://iridia.ulb.ac.be/~marchal/
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