Hi Brent, I have joined you last two posts,
Le 31-août-07, à 17:55, Brent Meeker a écrit :
>> Yes. I can accept that PA is a description of counting. But PA, per
>> se,
>> is not a description of PA. With your term: I can accept arithmetic is
>> a description of counting (and adding and multiplying), but I don't
>> accept that arithmetic (PA) is a description of arithmetic (PA). Only
>> partially so, and then this is not trivial at all to show (Godel did
>> that).
>
> Right. PA is description of arithmetic and Godel showed that part of
> PA could be described within PA.
OK. And then PA can reason on PA (without any new axioms).
>>> Do you see why I think your objection was a non-sequitur?
>>
>>
>> But then how do you distinguish arithmetic and Arithmetic? How to you
>> distinguish a description of counting (like PA or ZF) and a
>> description
>> of a description of counting, like when Godel represents arithmetic in
>> arithmetic, or principia mathematica *in* principia mathematica ...
>
> Yes, that's the question. If arithmetic is all that is provable from
> PA it is well defined.
Sure.
> But Arithmetic isn't well defined.
Why? It is just the set of arithmetical sentences which are satisfied
in the structure (N, +, *).
That set is not recursively enumerable (mechanically enumerable), but
non RE sets abound in math.
Do you accept that classical logic works on number? If yes, it is
simple to define Arithmetic in naive set theory, as you can define it
in formal set theory (ZF).
Recall that arithmetical truth (Arithmetic) is even just a tiny part of
mathematical truth. Analytical truth, second-order logic truth, etc...
are vastly bigger. Most usual categories are still bigger, ...
> It's lots of different sets of propositions that are provable from
> PA+Something.
Sure, and if that Something is sound, this gives sequences of
approximation of Arithmetic. Arithmetic, the set of true arithmetical
sentence is productive (like the set of growing functions I have
described to Tom). It means that not only Arithmetic is not recursively
enumerable, but it is constructively so! For each RE set W_i which is
propose as a candidate for an enumeration of Arithmetic, you can find
an element in Arithmetic (a true sentence) which does not belong to
W_i. This is a version of incompleteness.
> If that Something is a Godel numbering scheme then there is a mapping
> between proofs and arithmetic propositions.
I don't understand. The goal of a numbering scheme is to study what PA
can say about PA, without adding any "something" to PA.
>
>>
>> It is different. In Peano Arithmetic, a number like 7 is usually
>> represented by an expression like sssssss0 (or
>> s(s(s(s(s(s(s(s(0))))))).
>> But in arithmetical meta-arithmetic, although the number 7 is still
>> represented by sssssss0, the representing object "sssssss0" will be
>> itself represented by the godel number of the expression "sssssss0",
>> which will be usually a huge number like
>> (2^4)*(3^4)*(5^4)*(7^4)*(11^4)*(13^4)*(17^4)*(19^5), where 4 is the
>> godel number of the symbol "s" and 5 is the godel number of the symbol
>> "0". Prime numbers are used so that by Euclid's fundamental theorem,I
>> mean the uniqueness of prime decomposition of numbers (Euclid did not
>> get it completely I know) we can associate a unique number to finite
>> strings of symbols.
>>
>> In PA the symbols "0", "+" etc. make it possible to describe numbers
>> and counting. Meta-PA is a theory in which you have to describe proofs
>> and reasoning. Then it is not entirely obvious that a big part of
>> Meta-PA can be described in the language of PA.
>
> But this doesn't make it the same thing as PA.
It is functionnally isomorphe. Like a comp doppelganger. It is a third
person self. PA cannot prove that PA is PA, but can bet on it correctly
by chance. Then you can mathematically study what PA can prove on PA,
or what a (correct) machine can prove about herself, from some correct
third person description of herself. Don't confuse that third person
self with the first person whose "self" has not even a name.
>
>> This makes possible for
>> PA to reason about its own reasoning abilities, and indeed to discover
>> that "IF there is no number describing a proof of a falsity THEN I
>> cannot prove that fact", for example. This shows that wonderful thing
>> which is that PA can, by betting interrogatively on its own
>> consistency, infer its own limitation with respect to the eventually
>> never completely and effectively describable Arithmetic (arithmetical
>> truth).
>
> But it is this incompleteness and indescribability of Arithmetic which
> causes me to think that it doesn't exist.
"arithmetic" (PA) is incomplete (provably so by us, as far as we are
sound lobian machine), but it is describable, already by a simpler than
us machine like ZF. Then ZF cannot define a notion of set theoretical
truth.
It is general: no sound machine M can ever describe or define his own
global notion of M-truth. You can't infer from this that such notion
are senseless. All notion of M-truth can be define by some sound lobian
machine M' with M' far more complex than M.
> You are just betting on it. I look at it this way:
>
> objects = things we observe to exist.
'course a platonist stops here. He/she would say appearance = things we
observe.
I don't thing we ever see "existence" of something. That is always an
inference, *hopefully* stable. (unlike in night dreams for example).
> counting = a physical process associating objects
I can agree that when a human counts, there is some physicalness
involved. But counting has more general mathematical grounding. That is
what PA is about.
> arithmetic = propositions about counting
Yes, but Arithmetic too.
> Peano's axioms = a description of arithmetic
Nooooooo .... Just a little machine trying to scratch a tiny part of
it. (I guess you mean Arithmetic), because arithmetic is just the
theorems of PA: most of the time I identify a theory, with its set of
theorems. Like I say <>t is in G* but not in G ....
> PA+Godel = a description of proofs in PA
> meta-mathematics = description of PA+Godel and other math.
? Metamathematics makes sense only because it can be arithmetized.
Even the metamathematics of set theory.
> .
> .
> meta^N-mathematics = description of meta^(N-1)-mathematics
You can make the whole tower already in PA, or in richer theories if
you want more theorems, but then you climb just on the ignorance space
of mathematics. The interest of the arithmetization of arithmetic is
that you collapse the level with the meta-level.
>
> So, do I need Arithmetic?
Arithmetic is like qualia or pain. Perhaps we don't need that, but we
just cannot put that under the rug. Almost all of math would disappear
without Arithmetic.
>> Let me sum up by singling out three things which we should not be
>> confused:
>>
>
> OK. I don't think I'm confused about them.
>
>>
>> 1) A theory about numbers/machines, like PA, ZF or any lobian machine.
>> (= finite object, or mechanically enumerable objet)
>>
>> 2) Arithmetical truth (including truth about machine). (infinite and
>> complex non mechanically enumerable object)
>
> Is this the set of all (countably infinite) true propositions about
> the natural numbers? Is the existence of this set a matter of faith?
For a machine like PA, who happens to talk only on numbers, yes, it is
a matter of faith for him/she/it. But for a lobian machine like ZF, the
esistence of Arithmetic is a matter of proof.
>> Only a meta-theory *about* PA, can distinguish PA and arithmetical
>> truth. But then Godel showed that sometimes a meta-theory can be
>> translated in or by the theory/machine. Rich theories/machine have
>> indeed self-referential abilities, making it possible for them to
>> guess
>> their limitations.
>
> Are there not infinitely many Godel numbering schemes.
Those defining PA in PA will be equivalent, unless they describe
intensionnal variants like the hypostases, but then that is really new
non equivalent points of view. Usually it is said that the choice of
the numbering does not matter.
>
>> By doing so, such machines infer the existence of
>> something transcendenting <transcending> (if I can say) themselves.
>>
>> OK?
>
> OK.
All right,
Bruno Marchal
http://iridia.ulb.ac.be/~marchal/
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Received on Sun Sep 02 2007 - 10:47:55 PDT