Are there any large cardinals that are not ordinals?

Question

In ZF, are there any useful large cardinal that cannot be well-ordered? I think that some of the partition cardinals are that way, since with AC, we cannot have $\kappa \to (\omega)^{\omega}$. Are there any other cases like this?

Answer 1

The question is what do you mean by large cardinals and what do you mean by useful?

For example, it is consistent that there exists a set $A$ such that the cofinite filter on $A$ is an ultrafilter. We can even show that such ultrafilter is closed under intersection of ordinal-indexed sequences (mainly because under this assumption every such sequence is actually finite).

But what happens when we try to take an ultrapower of the universe with this $A$ and this ultrafilter? Well, it turns out that every function $A$ into the ordinals has a finite range. So every function to the ordinals is constant on a cofinite set. We can show, if so, that the ultrapower embedding is in fact the identity from this.

So a set which is "like a measurable" can exist, but is it useful in the same way? Not really. It is not to say that these sets are entirely useless. Eilon Bilinski wrote his masters thesis about using such sets in order to sort of mimic Prikry forcing and obtain all sort of interesting choiceless results. But it is certainly not the same thing.

Similarly we can get sets which are going to satisfy all sort of strange Ramsey properties, but not because they are "large" in any sense, but because of some silly reason like here. They don't have enough subsets, or something like this. Which in turn means that we cannot use them like we would use normal large cardinals.

Generally speaking we can get results which state that the continuum is "large" in some sense. For example $\sf AD$ has all sort of interesting implications about the continuum which say that it is somewhat large, but one might argue that these things also come from the fact that there are plenty of ordinals which are large in some good sense (for example, the club filter on $\omega_1$ is an ultrafilter).

I hope this answer helped to dispel some of the confusion and create a new one regarding what does it mean for something to be useful, or large. With or without choice.