[FOM] Mac lane set theory

[Jeremy Bem]

I was assuming one would try to substitute "the Von Neumann universe"
for M, but perhaps you're proposing "a set of inaccessible
cardinality".  My understanding is that the former concept is felt to
justify modern set theory in some way, even though it doesn't seem to
be rigorous and certainly doesn't describe a set.  Presumably the
latter concept doesn't seem sufficiently constructive to play this
justificatory role.  Alice's M combines the features of both.  So
there is a real difference...no?

-Jeremy

On Tue, Dec 22, 2009 at 12:48 PM, rgheck <rgheck at brown.edu> wrote:
> 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
>
>
>