[FOM] reducing higher to second order?

[Marcus Rossberg]

On Sunday, December 21, 2003 4:33 AM, H. Enderton wrote:
> Todd Wilson wrote:
> >I have seen passing references in the literature to a reduction of
> >higher-order logic to second-order logic, but ...
>
> I think of that as Montague's result, but Vaught and (independently)
> Hintikka might be mentioned.
>
> Lemma:  In second-order logic, you can express "A is the power set
> of B" (using a binary predicate symbol to simulate epsilon).
>
> Consequence:  The set of validities of k-order logic is recursively
> isomorphic to the set of second-order validities.
~snip~

It might be useful to emphasis the following limitations:
(1) This result only applies to validities (as stated above by Professor
Enderton), not to the consequence relation in general.
(2) The proof relies on the domain being set-sized.

See Stewart Shapiro's _Foundations Without Foundationalism_, pp. 137--141.

Best wishes,
Marcus Rossberg


---------------------------------------------------------------
Marcus Rossberg
Arché - The AHRB Research Centre for the
Philosophy of Logic, Language, Mathematics and Mind
University of St Andrews
St Andrews, Fife KY16 9AL
Scotland, U.K
---------------------------------------------------------------