Universe cardinals and models for ZFC

Question

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal $\gamma$ such that $V_\gamma$ models ZFC. For example, if an inaccessible cardinal $\kappa$ exists, then $\kappa$ is a universe cardinal. But furthermore, by the Löwenheim-Skolem theorem, there are then lots of models of ZFC inside $V_\kappa$, and Hamkins says that this proves that $\kappa$ is the $\kappa$th universe cardinal. Specifically, $\{\gamma < \kappa: V_\gamma \prec V_\kappa\}$ is evidently closed in $\kappa$, and is "unbounded by Löwenheim-Skolem".

1. This is confusing to me. The Löwenheim-Skolem theorem says that $V_\kappa$ contains submodels of arbitrary cardinality. But in order to have any universe cardinals at all, you need models of ZFC of the form $V_\gamma$, which don't seem to come from Löwenheim-Skolem. Why should these exist? Is there something you can do to a model to put it in the form $V_\gamma$?

2. Relatedly, it's also true (this is in Kanamori's Higher Infinite) that the only models for second-order ZFC are $V_\kappa$ for $\kappa$ inaccessible. Can someone explain the difference between these two concepts? I'm finding the distinction between first-order and second-order Replacement a little hair-splitting.

I hope I've phrased this the right way -- I'm a set theory rookie. Thanks in advance!

Answer 1

Yes, you do need a tiny additional argument to get universe cardinals below an inaccessible $\kappa$. The additional step is an elementary chain argument, which tends to occur relatively frequently.

Starting from an inaccessible $\kappa$, Löwenheim-Skolem gives a small elementary submodel $M_0\prec V_\kappa$. We can then put $M_0$ inside $V_{\alpha_0}$ for some cardinal $\alpha_0<\kappa$. Applying Löwenheim-Skolem again gives a new elementary submodel $V_{\alpha_0}\subseteq M_1\prec V_\kappa$. We then put $M_1$ into some $V_{\alpha_1}$ again. After repeating this $\omega$ many times we get intertwined chains of models $M_n$ of ZFC and initial segments $V_{\alpha_n}$ of the universe which both union up to $V_{\sup \alpha_n}$, which is elementary in $V_{\kappa}$. This implies that $\sup\alpha_n<\kappa$ is a universe cardinal.

By the way, the terminology has shifted slightly since Joel wrote those notes and the term now used for these cardinals is worldly cardinals.

Answer 2

Miha gave a very good answer to the first question. Let me add a bit on the second question.

The difference between first-order and second-order replacement is subtle. First-order replacement is a schema, and it says that if $f$ is an internally definable function with a set domain, then its range is a set; whereas the second-order replacement axiom is a single axiom, and it states that whenever $f$ is a subset of $V_\kappa$ which is a function, and its domain is a set of $V_\kappa$, then its range is an element of $V_\kappa$ as well.

If $V_\kappa$ satisfies the second-order axiom, then it $\kappa$ to be regular. To see this note that:

1. If $A\subseteq\kappa$ is an unbounded set, then $A\notin V_\kappa$.
2. Every unbounded set of ordinals is the range of a function which enumerates it.

Now if $\kappa$ is singular, there is some $\gamma<\kappa$, and a function $f\colon\gamma\to\kappa$ which enumerates an unbounded set. This function $f$ is a subset of $V_\kappa$, and its domain is an element of $V_\kappa$, but its range is not.