FOM: sterility?
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Tue Mar 23 02:22:48 EST 1999
> Date: Mon, 22 Mar 1999 16:45:50 -0700
> From: Randall Holmes <holmes at catseye.idbsu.edu>
>Second-order logic allows <...> a categorical axiomatization
>of N and other familiar structures;
>first order logic, even formalization in a first-order
>theory as strong as ZFC, does not.
a) 2nd order logic axiomatizes N caregorically
through appeal to subsets of N.
b) 1st order logic does it with equal
elegancy: simply require, in addition to
the 1st order Peano axioms, that every
proper nonempty initial segment has a
maximal element.
Is there any clear, well defined reason why
2nd logicians consider a) as so remarkably
superior to b) as to make it the sole
cornerstone of a competitive system of f.o.m. ?
Apparently, the only mathematical point here is that
some properties of mathematical structures S,
not expressible in the *corresponding* 1st-order
language,
become expressible if quantification
over P(S) is allowed.
V.Kanovei
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