[FOM] second order ZFC

Allen Hazen allenph at unimelb.edu.au
Wed Mar 19 02:53:55 EDT 2008

Charles Parsons has mentioned

> 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.

I'd like to endorse his judgment that
> That result is of philosophical interest, although it's not easy to
> say what its significance is.

and his (implicit) recommendation of
>Vann McGee, "How we learn mathematical language," _Philosophical
> Review_ 106 (1997),
for an interesting philosophical discussion.

And I'd like to ADD a bibliographical item.  Zermelo's work is discussed
(and claims made for its philosophical significance) in (particularly the
final part of) Marcus Giaquinto's book "The Search for Certainty" (Oxford:
Clarendon Press, 2002).

(I ***think*** that the Axiom of Replacement is essential here: as I
remember it, Gabriel Uzquiano showed that the class of models of
second-order Zermelo set theory is not as nicely constrained in his "Models
of second-order Zermelo set theory," in "Bulletin of Symbolic Logic" vol. 5
(1999), pp. 289-302.)


Allen Hazen
Philosophy unit
University of Melbourne

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