Le 19-août-06, à 08:48, Brent Meeker wrote quoting Stathis Papaioannou

>>
>> What more could we possibly ask of a theorem other than that it be 
>> true relative to some
>> axioms? That a theorem should describe some aspect of the real world, 
>> or that it should
>> be discovered by some mathematician, is contingent on the nature of 
>> the real world, but that
>> it is true is not.
>
> That it is a true description of the real world, or that it is a true 
> theorem
> relative to the axioms.  It is a mistake to conflate the two, which I 
> suspect is
> done by people claiming mathematical theorems are true.


No. It is done by people claiming true mathematical propositions are 
theorem.

Robinson Arithmetic (Q or RA) and Peano Arithmetic (PA), which in our 
context are better seen as a (mathematical) *machines*, are SOUND with 
respect to the so-called (by logicians) standard model of arithmetic, 
which is the mathematical structure (N,+,*) given by the non negative 
integers N together with addition and multiplication (learned in high 
school).
Now RA and all its consistent extensions (and thus PA, "ZF", ...) are 
INCOMPLETE with respect to that mathematical structure (N,+,*), in the 
sense that for any of those theories there exist always infinitely many 
true propositions, "true" meaning really: satisfied by (N,+,*) which 
are unprovable by those theories.
There is no complete TOE for the "standard" additive and multiplicative 
behavior of the natural numbers.
But there is nothing wrong asserting that a theorem of PA is true 
(always with that meaning of being statisfied in (N,+,*)), because 
nobody (serious) doubt the axioms of PA, or doubt truth couldn't be 
preserved by the modus ponens inference rule or by the quantifier rules 
(and thus nobody doubts in the theorems proved by PA).

Bruno

http://iridia.ulb.ac.be/~marchal/


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