Are all large cardinal axioms expressible in terms of elementary embeddings?

Question

An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.

A critical point of such an embedding is the least ordinal $\alpha$ such that $f(\alpha)\neq\alpha$.

A large cardinal is a cardinal number that cannot be proven to exist within ZFC. They often appear to be critical points of an elementary embedding of models of ZFC where $M$ is the von Neumann hierarchy, and $N$ is some transitive model. Is this in fact true for all large cardinal axioms?

Answer 1

No. If $\kappa$ is the critical point of a full elementary embedding $j : V\to N$, with $N$ transitive, then $\kappa$ is measurable. Yet not all large cardinals are measurable; see, for instance, weakly compact cardinals. Even large cardinals with higher consistency strength (like Woodin cardinals) are not always measurable.

Answer 2

Not exactly.

First of all, there are small large cardinals, such as inaccessible or Mahlo cardinals, for which I do not know of any natural formulation in terms of embeddings.

Once we reach weakly compact cardinals, we can start expressing traditional large cardinal properties in terms of embeddings, but the models involved only satisfy fragments of $\mathsf{ZFC}$.

In the realm of large cardinals past measurability, it is true that the large cardinal template is often used, but you need to be careful. Woodin cardinals, for example, are not even measurable in general (though they are the limit of measurable cardinals).

In the choiceless context, one studies for example partition cardinals, and the natural formulation of these axioms is not in terms of embeddings; in fact, even though the partition properties typically imply measurability, I do not know of an embedding formulation that fully captures strong partition cardinals.

Finally, there are large cardinal notions that do not come associated with a large cardinal per se, such as the existence of $0^\sharp$, and even though the relevant sets provide us with embeddings, these embeddings are typically only of partial (thin or set sized) models.

You may want to visit Cantor's attic for an overview of the main signposts in the large cardinal hierarchy.

Answer 3

The existence of embedding characterizations of many large cardinal notions is something of a happy surprise really. There seems to be no reason, a priori, for the large cardinals of a more combinatorial nature to have such a description, but here we are.

Still, if you require the domain of your embedding to be the whole universe, your cardinal will always be at least measurable. To get weaker large cardinals, you need to allow smaller structures as domains. Also, you will usually need many such embeddings to verify your large cardinal property (but this is true for higher large cardinals as well, e.g. supercompact cardinals).

As a prime example, consider weakly compact cardinals. These have an embedding characterization saying that $\kappa$ is weakly compact if there is an embedding with critical point $\kappa$ from every transitive set of size $\kappa$ and containing $\kappa$.

I recommend you take a look at Cantor's Attic, where many large cardinals are described along with their various characterizations.

Answer 4

There is an online PDF of lecture slides by Woodin on the "Omega Conjecture" in which he axiomatizes the type of formulas that are large cardinals. I do not know how exhaustive his formulation is. See the references under

http://en.wikipedia.org/wiki/Omega_conjecture