[FOM] second order ZFC

Charles Parsons parsons2 at fas.harvard.edu
Mon Mar 17 17:02:19 EDT 2008

At 10:46 PM +0100 3/5/08, pax0 at seznam.cz wrote:
>Is anybody familiar with second order set theory ZFC to 
>explain/point to a paper
>what language it uses, what axioms and models it has,give a typical 
>theorem, or perhaps
>indicate its philosophical value.
>Thank you, J.P.

There is a classic paper by Zermelo, "Ueber Grenzzahlen und 
Mengenbereiche," _Fundamenta Mathematicae_ 16 (1930), which in effect 
characterizes the standard models of second-order ZF. They are (up to 
isomorphism) the initial segments of the cumulative hierarchy up to 
some strongly inaccessible cardinal. (Some would add as a "class 
model" the whole universe, but Zermelo did not.)  It follows that 
given any two such models, either they are isomorphic or one is 
isomorphic to the sets of rank < k, for some strong inaccessible k in 
the other.

That result is of philosophical interest, although it's not easy to 
say what its significance is. Two rather different applications of it 
are Vann McGee, "How we learn mathematical language," _Philosophical 
Review_ 106 (1997), and D. A. Martin, "Multiple universes of sets and 
indeterminate truth-values," _Topoi_ 20 (2001).

Zermelo allows urelements, which of course makes the isomorphism type 
also depend on the number of urelements.

There is an English translation of Zermelo's paper in volume 2 of 
William Ewald (ed.), _From Kant to Hilbert_ (Oxford 1996).

About Peter Koellner's work, Marcus Rossberg may have in mind his 
paper "Strong logics of first and second order," forthcoming in the 
BSL. It doesn't deal directly with second-order ZFC, but it does 
point out that standard validity in second-order logic is not robust 
under forcing, even with large cardinal assumptions (in contrast, in 
particular, to Woodin's omega-logic). That is relevant to the appeal 
to second-order logic in discussions of the continum hypothesis.

Charles Parsons

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