Separation of large cardinal notions

Kunen came up with the idea of, starting with a large cardinal notion A, kill the weak compactness of A and then resurrect some property without resurrecting weak compactness. Here we can for instance start with a weakly compact and find a generic extension in which the cardinal reflects stationary subsets but isn't weakly compact. Other examples include starting with a Ramsey there's an extension where it's virtually Ramsey but not Ramsey, and starting with a stationary cardinal there is an extension where it admits stationary reflection but isn't stationary. The specific definitions of these large cardinals is not important to my question, just that weakly compact => stationary => admits stationary reflection and Ramsey => virtually Ramsey.

My question is then: what does this really show? It seems that we can only produce an upper bound for the consistency of the separation. Does this imply that the two large cardinal notions in question are really different? And if yes, in what sense? It seems to me that statements of the form

Con(ZFC + kappa is A) => Con(ZFC + kappa is A + there is a lambda which is B but not C),

would give the separation, as then we get that

ZFC+"there exists an A" doesn't prove that "every B is C".

Most statements I've found isn't of this form however. I know my question is vague, but hopefully it at least makes sense.

• Kunen shows that ZFC doesn't prove "If $\kappa$ reflects stationary subsets, then $\kappa$ is weakly compact." So the large cardinal notions "weakly compact" and "reflects stationary subsets" are different, unless each is inconsistent. I'm a bit confused; what more are you hoping for? – Noah Schweber Jul 15 '17 at 17:30
• Maybe I'm simply a bit confused, which is highly plausible. Because he shows what you state, assuming there's a weakly compact in V, right? I suppose all I'm asking is instead of this difference in theories why don't we require that such separation has to be of the kind 'ZFC+"there is a weakly compact" doesn't prove that "every cardinal admitting stationary reflection is weakly compact"'? In other words, that there is a weakly compact cardinal in the forcing extension? – Dan Saattrup Nielsen Jul 15 '17 at 19:57

1 Answer

Let me argue that Kunen's argument actually shows the best possible thing here.

First, let's think about consistency results. The "ideal" result here would be:

(i) Con(ZFC) implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal.")

Obviously we can't prove this in any reasonable theory (I assume that we try to prove consistency results in something vastly weaker than ZFC) unless ZFC is inconsistent, so the actual ideal result is:

(ii) Con(ZFC + "There is a stationary-reflecting cardinal") implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal").

Note that this is a strictly stronger statement than "Con(ZFC + "There is a weakly compact cardinal") implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal").

Although Kunen seems to begin with the assumption of the consistency of weak compactness, he's actually proving the ideal result (ii) above! This is because - assuming the consistency of stationary-reflecting cardinals - there are three cases:

• Stationary reflecting cardinals are inconsistent. Done, vacuously.

• Weakly compact cardinals are consistent. Kunen then builds a model with a stationary-reflecting, non-weakly-compact cardinal.

• Weakly compact cardinals are inconsistent, but stationary-reflecting cardinals are consistent. In this case we're also vacuously done.

The only interesting case is when everything involved is consistent, and that's what Kunen treats. So the apparent use of an additional consistency hypothesis is in fact unnecessary, since it only ignores cases (i) and (iii) which are each vacuous.

Now what about actual models?

Obviously if T is stronger than ZFC, then "T doesn't imply "every B is C"" is stronger than "ZFC doesn't imply "every B is C."" So we can ask, e.g.,

• Does ZFC + [large cardinal] imply that every stationary-reflecting cardinal is weakly compact?

The answer to this is basically no, since basically every large cardinal notion is preserved by small forcing. Start with a model with your large cardinal property holding above a weakly compact - say, $V\models$ "$\kappa$ is weakly compact and there is a supercompacts $>\kappa$." Now apply Kunen's construction; this is a small forcing relative to the supercompact, and so we get a model of "There is a supercompact and there is a stationary-reflecting non-weakly-compact cardinal."

This result essentially says that reasonable large cardinals tend to have no bearing on the implications between various combinatorial properties, since these implications tend to either hold or be breakable by small forcings. So to me this suggests that this isn't really the right place to look.

This ignores, of course, those large cardinal hypotheses which do affect combinatorics, like non-measurable weakly measurable cardinals, but these are a bit odd; similarly, I would argue that generic large cardinal hypotheses a la Foreman aren't really large cardinal hypotheses.

• Thanks a lot Noah, that really cleared things up! Also, technically speaking, ZFC+stationary-reflecting could prove that there is a non-weakly compact stationary-reflecting cardinal. In this case though we also get Jensen's result that every stationary-reflecting is weakly compact in L, so that the statement "every stationary-reflecting is weakly compact" is then independent of "ZFC+stationary-reflecting". I suppose that since this last direction relied on L, we can't rule out the scenario where ZFC+large cardinals proves that there is a non-weakly compact stationary-reflecting, right? – Dan Saattrup Nielsen Jul 16 '17 at 4:50