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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Received on Sat Aug 19 2006 - 08:51:47 PDT