[FOM] Mac lane set theory

rgheck rgheck at brown.edu
Tue Dec 22 12:48:54 EST 2009

On 12/21/2009 07:20 PM, Jeremy Bem wrote:
> For me, the appeal of weaker set theories is the possibility of
> incorporating them into a straightforward worldview such as the
> following:
> "Alice is a Platonist.  She believes in the objective reality of sets,
> and that first-order statements in the language of set theory have
> definite truth values.  She believes that the axioms of ZC (Zermelo
> set theory with choice) are true, and so of course she believes that
> ZC is consistent.  She also believes that by putting E := the empty
> set,  S := E U Pow(E) U Pow(Pow(E)) U …,  M := S U Pow(S) U
> Pow(Pow(S)) U …, she obtains a set which is a model of ZC.  This
> confirms her belief that ZC is consistent; in fact, it is for her an
> (informal) proof of the consistency of ZC.  She believes (and can
> prove formally) the fundamental theorem of algebra, Goodstein's
> theorem, and many other standard, beautiful results.  She doesn't know
> the truth value of Borel determinacy, projective determinacy, or the
> continuum hypothesis."
> Is there an objection to this worldview, and is there a similarly
> straightforward worldview based on ZFC?  In particular, it doesn't
> seem that one could get the same reassurance that Alice gets from M,
> from the definition of the Von Neumann universe.  I've never been at
> peace with that definition, but in particular, V isn't a set, is it?
> So it isn't a model in the usual sense of first-order logic.  This
> seems like a real sacrifice, not ad hoc, and not worth making just for
> Borel determinacy.
I'm not unsympathetic, but it's a little unclear to me precisely where 
the difference is supposed to lie between the construction Alice 
undertakes and one that could similarly be undertaken for ZFC. Alice's 
construction depends, obviously, upon resources that are not available 
in ZC, in particular, upon one instance of the axiom of replacement or, 
if you prefer, upon something like the ability to take countable unions. 
There are constructions one can undertake in ZFC plus a teeny bit that 
lead to a model of ZFC, too.

Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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