Le 30-août-07, à 20:21, Brent Meeker a écrit :
>
> Bruno Marchal wrote:
>>
>> ? I don't understand. Arithmetic is about number. Meta-arithmetic is
>> about theories on numbers. That is very different.
>
> Yes, I understand that. But ISTM the argument went sort of like this:
> I say arithmetic is a description of counting, abstracted from
> particular instances of counting.
I guess you mean by "arithmetic" some theory, i.e. Peano Arithmetic
(PA). I can agree that a theory like PA is a description of counting.
But Peano Arithmetic is different from Arithmetical truth. OK?
> You say, no, description of arithmetic is meta-mathematics
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).
> and that's only a small part of mathematics, therefore arithmetic
> can't be a description.
PA can be considered as a description of counting. But PA is incomplete
with respect to Arithmetic (with a big "A").
PA, like ZF, like any effective theory, (like ourselves with comp) can
only describe a tiny part of Arithmetic.
>
> 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 ...
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. 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).
So, no, I don't see why you think my objection is a non-sequitur. It
seems to me you are confusing arithmetic and Arithmetic, or a theory
with his intended model.
Bruno
PS The "beginners" should no worry. Most of what I say here will be
re-explain, normally. Just be patient (or ask, or consult my work with
some good book on logic).
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Aug 31 2007 - 05:00:56 PDT