[FOM] SOL vs. ZFC

[Joe Shipman]

***Surely the soundness of all commonly used systems of axioms for second-order logic is provable in ZFC?***

I thought there might be a version with the strength of a second order set theory like the Morse-Kelley system.

I am motivated by the question "how much of mathematics is logic in disguise?". If second order logic in the standard formalism is really "logic", then logic can express almost all of the mathematics anyone cares about, and the question reduces to which axioms can plausibly be described as "logical". 

I found a good paper by J. Vaananen which deals with such matters:

http://www.math.helsinki.fi/logic/people/jouko.vaananen/solost.pdf

The conclusion I would draw is that if you don't regard ontological questions as important and only care about statements of the form "structures satisfying X also satisfy Y", then any theorem of "ordinary mathematics" that can be proven from Maclane Set Theory can be defended as "logic".

I know of no theorems of "ordinary mathematics" that follow from Zermelo Set Theory but not Maclane Set Theory, can anyone suggest one?

McLarty, in a paper "Set Theory for Grothendieck's Number Theory"

http://www.cwru.edu/artsci/phil/Groth%20found26.pdf

suggests that V_w•3 suffices for any application of the framework developed by Grothendieck, which originally used a "Universes" Axiom equivalent to a proper class of inaccessibles. I'd hesitate to justify results derived from "V_w•3 exists" as purely logical, but Logicism gets pretty far and I wonder why it seems to be unpopular.

-- JS
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