[FOM] First Order Logic

[Carl Hewitt]

I am having trouble understanding why the proponents of first-order logic think that second-order systems are unusable.

[Dedekind 1888] and [Peano 1889] thought they had achieved success because they had presented axioms for natural numbers and real numbers such that models of these axioms are unique up to isomorphism with a unique isomorphism.  And later generations of mathematicians were happy to use these axioms.

The above axiomatizations bar many mathematical monsters that are created by the "first-order thesis" [Barwise 1985].  The article http://arxiv.org/abs/0812.4852 presents a mathematical foundation for Computer Science that is an alternative to the first-order thesis.

Regards,
Carl

From: MartDowd at aol.com
Sent: Wednesday, August 28

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).

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