Re: A question about FOL theories and models



Nam Nguyen wrote:


...(Think of Shoenfield's
finite axiomatization of N, and the "standard model" of arithmetic
of the natural numbers, which is N itself - which is quite circular!
[Given that we're talking about the foundation of reasoning!])


Imho, it's time that we need to revamp FOL, to move on with modern
day needs of reasoning. We should not pretend, for instance, that
the arithmetic syntactic formalization is consistent, while we just
*assume as a priori* the existence of a "model", without knowing
for absolute logical certainty that this "precious" one be indeed
a model.

Consider, for example, the possible situation in which ~GC is
arithmetically (interpreted to be) true, while GC is *also provable*
in Shoenfield's 'N' formal system. The potential _peril_ of the current
FOL is that we've not *absolutely* excluded this possibility!
(Needless to say, neither have we proven absolutely that (GC /\ ~GC)
is not provable in PA: we've only shown it's - relatively - so).

--
-----------------------------------------------------

What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
----------------------------------------------------

.