[FOM] Categoricity in f.o.m.

[Joe Shipman]

This shows an alternate history FOM could have taken. If you simplify Godel's construction of L slightly you get the Cohen/Shepherdson minimal model M you describe here, and the rest of Godel's 1938 proof of the consistency of AC and GCH goes through in the same way. Cohen called these the "strongly constructible sets" and it's not an unreasonable strengthening of V=L to assume "only those sets which must exist do exist". Godel's L only applies this restriction "horizontally" but you might as well do it vertically too and then you get true categoricity, and the axiom has a simpler logical type than V=L because it can be stated in the form "there is no countable structure of a certain kind".

-- JS

Sent from my iPhone

> On Apr 1, 2016, at 3:26 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
> 
> John Baez on March 31, 2016 9:10 PM
> https://golem.ph.utexas.edu/category/2016/03/foundations_of_mathematics.html
> wrote
> 
> "I believe many people who like set-theoretic foundations still
> secretly wish for axioms that are categorical — in a sense that has
> little to do with category theory. Axioms are ‘categorical’ if all
> models of those axioms are isomorphic: that is, roughly, the axioms
> uniquely determine the structure they’re axiomatizing.
> 
> Of course, we know that no reasonably powerful axioms of set theory
> can be categorical. Nonetheless some people still speak of ‘the’
> universe of sets, as if there were some unique structure that we are
> seeking to axiomatize. They also argue, at times, about which axioms
> are ‘true’.
> 
> In geometry, this attitude was made obsolete long ago by the discovery
> of non-Euclidean geometries. But in set theory, despite Gödel, it
> persists.
> 
> Thanks to the ‘modular’ approach you describe, category theorists have
> a very different attitude. We are happy with axioms that have many
> dramatically different models. If we speak of ‘a category with finite
> products’, or ‘a topos with natural numbers object’, we consider it a
> feature, not a bug, that these axioms apply to many different
> structures.
> 
> We are not trying to axiomatize ‘the universe’. We are trying to
> specify a kind of context in which one can do a certain class of
> interesting things."
> 
> ACTUALLY, I had meant to make an FOM posting on Categoricity in Standard f.o.m.
> 
> Categoricity does play a GREAT ROLE in standard f.o.m. But it is
> important to distinguish some different kinds of categoricity, some of
> which are impossible as Baez suggests, and others being principal
> hallmarks of standard f.o.m.
> 
> Baez is quite right in the sense that ZFC is not categorical in the
> usual sense. This means that any two models of ZFC are isomorphic.
> False.
> 
> However, there is still a lot of very important categoricity to be had
> with models of ZFC. Let me place the following condition on models of
> ZFC.
> 
> M RESPECTS INCLUSION. Let x be a point in the model M of ZFC. Let y be
> an actual set consisting of only elements of x in the sense of M. Then
> there exists a point x* in the model M whose elements in the sense of
> M are exactly the elements of y.
> 
> THEOREM 1. Among the inclusion respecting models of ZFC, there is a
> unique one, up to isomorphism, which embeds in every such. This is the
> so called minimum inclusion respecting model of ZFC.
> 
> THEOREM 2. Any two inclusion respecting models of ZFC + "there is no
> inclusion respecting model of ZFC" are isomorphic.
> 
> There are other theorems like this, also for NBG.
> 
> The categoricity properties of finite set theory are much stronger.
> 
> Harvey Friedman
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom