Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$.

Large cardinal properties generally come in one of three flavors - combinatorial (e.g. Mahlo), logical (e.g. indescribability), and model-theoretic (e.g. measurable). This last class has turned out to be in many ways the most fundamental - certainly, the model-theoretic large cardinal properties are the strongest (in terms of both consistency strength, straightforward logical strength, and - usually - size) by far. The prototypical model-theoretic large cardinal axiom is the following formulation of a measurable cardinal:

There is a nontrivial elementary embedding of $V$ into some inner model $M$.

The other model-theoretic large cardinal axioms assert the existence of elementary embeddings with additional nice properties - they "cohere" nicely, their codomains are large, etc.

I'm curious whether a similar phenomenon is known to exist with regard to some other set theory - let's take $NFU$ as the most likely example. It is not at all obvious to me that the existence of a nontrivial elementary embedding of $V$ into some inner model $M$ has any strength at all over $NFU$ (and in fact, it's not totally obvious to me what an "inner model" should even be!). So my question is: are there axioms which, when added to $NFU$, result in high consistency strength (say, at least that of $ZFC$ + a measurable), which are "model-theoretic" in flavor? Of course, this last requirement is very subjective; I'm hoping that, nonetheless, this question is meaningful.

I'm especially interested in model-theoretic large cardinal axioms over $NFU$ which seem not to have obvious $ZFC$ counterparts - that is, we can show that $NFU+(*)$ is consistent relative to large cardinals, and has consistency strength at least that of a measurable, but we can't pin down the exact strength, nor is there an obvious guess as to what it should be.

To head off triviality, let's ignore axioms which are the relativization of a $ZFC$ large cardinal property to the Cantorian part of a model of $NFU$. :)

• A pathway to the answers you seek are in the "Strong axioms of infinity" section of (of all places) the Wikipedia entry "New Foundations". The author of this entry claims that these results are found in some unpublished work of Solovay. It would be nice if Prof. Solovay would provide us access to this unpublished work so we could check this out for ourselves.... – Thomas Benjamin Oct 11 '15 at 23:21
• @ThomasBenjamin I read that section; however, if I am reading it correctly, those axioms don't climb past a measurable, which is what I ask for. (I guess this relies on my vague recollection that "'MK + 'Ord is measurable'' is consistent relative to 'ZFC + a measurable'" is correct, though.) – Noah Schweber Oct 11 '15 at 23:26
• Though you mention you are especially interested in model-theoretic large cardinal axioms over $NFU$ ($NFU$ being, according to your question ,"the most likely example" of an alternative set theory where such model-theoretic large cardinal axioms 'naturally hold' in the sense of $NFU$+ "Infinity"+"Large Ordinals"+Small Ordinals" being equivalent $MK$+"the proper class ordinal is a measurable cardinal"), you mention you would consider "set theories other than $ZFC$". Consider the set theories found in Harvey Friedman's preprint – Thomas Benjamin Oct 15 '15 at 10:37
• (cont.) "Axiomatization of Set Theory by Extensionality, Separation, and Reducibility" (found in the "downloadable" section of his Homepage). Would these set theories found in this preprint be sufficiently 'alternate' to constitute an answer to your question? – Thomas Benjamin Oct 15 '15 at 10:43
• @ThomasBenjamin That's a really interesting paper - thanks for the reference! It's still pretty close to ZF, though. I'm ultimately looking for something further from ZF. – Noah Schweber Oct 18 '15 at 23:33