[FOM] First Order Logic
MartDowd at aol.com
MartDowd at aol.com
Wed Aug 28 12:06:02 EDT 2013
Expanding on this point, the difference between first-order and
second-order logic is that in second-order logic set variables are required to range
over subsets of the domain. Second-order logic is not usable in practice
because there is no recursively enumerable set of axioms (theorem 41C of
Enderton's logic book).
Second-order set theories (such as NBG) are actually first-order theories
because they are given as a set of axioms, and proofs must use the rules of
(many-sorted) first-order logic. In both category theory and small large
cardinal theory, this is a very useful system, and is a conservative
extension of ZFC I think logicians will always be interested in whether
theorems of mathematics can be proved in ZFC. We already know from Boolean
relation theory that this is not always the case.
- Martin Dowd
In a message dated 8/26/2013 7:56:59 A.M. Pacific Daylight Time,
hmflogic at gmail.com writes:
Imagine the situation if one proposed a second order version of ZFC, in
the serious sense of second order here (not the fake notion used in so called
second order arithmetic, where it is simply a two sorted first order
theory). There would be widespread disagreement about whether a purported proof
within second order ZFC is correct or not. This directly stems from the fact
that it is not a first order formulation.
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