[FOM] Categoricity in f.o.m.

[Harvey Friedman]

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