Bruno Marchal wrote:
> 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).

Right.  PA is description of arithmetic and Godel showed that part of PA could 
be described within PA.

>> 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 ...

Yes, that's the question.  If arithmetic is all that is provable from PA it is 
well defined.  But Arithmetic isn't well defined.  It's lots of different sets 
of propositions that are provable from PA+Something.  If that Something is a 
Godel numbering scheme then there is a mapping between proofs and arithmetic 

> 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.

>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.  You are just betting on it.  I look at it 
this way:  

objects = things we observe to exist.
counting = a physical process associating objects
arithmetic = propositions about counting
Peano's axioms = a description of arithmetic
PA+Godel = a description of proofs in PA 
meta-mathematics = description of PA+Godel and other math.
meta^N-mathematics = description of meta^(N-1)-mathematics

So, do I need Arithmetic?

Brent Meeker

> 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).
> > 

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