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 

> 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 
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 

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.


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).


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