It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are:

Question (1): Are there any large cardinals for which this result is unknown?

Question (2): In particular is the above result true for some small large cardinals like "reflecting cardinals", "unfoldable cardinals", "ineffable cardinals", "subtle cardinals" and ...?

Question (3): Is there any large cardinals whose consistency with $GCH$ is not known?

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    $\begingroup$ Related. $\endgroup$ – Andrés E. Caicedo Jul 22 '13 at 15:04
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    $\begingroup$ As an aside, let me remark that in the second part of his Notices paper, e-math1.ams.org/notices/200107/fea-woodin.pdf, Woodin deduces from his $\Omega$-conjecture that all large cardinal axioms are well ordered. Obviously, a general definition for large cardinal axioms is required for such a claim. Now the above property (resistance under small forcing) is included into Woodin's definition. $\endgroup$ – Péter Komjáth Jul 23 '13 at 12:44

These are sweeping questions, whose full answer would constitute several book chapters. So let me just sketch some general information. (If you want, please ask a more focussed question about specific notions.)

The basic situation is that the Levy-Solovay phenomenon is extremely widespread, and holds for nearly all of the usual large cardinal notions. For large cardinal properties witnessed by embeddings $j:V\to M$ with critical point $\kappa$, where the forcing has size less than $\kappa$, it is easy to see that the embedding lifts to the forcing extension $j:V[G]\to M[j(G)]$ and still generally witnesses the large cardinal property in the extension. This kind of reasoning applies to measurable cardinals, supercompact, partially supercompact, strongly compact, strong, huge, almost huge, etc. etc. By considering smaller embeddings $j:M\to N$, the same idea works with weak compactness, unfoldability, strong unfoldability, uplifting cardinals, extendible cardinals, etc. etc. If one lifts $V_\kappa\prec V_\theta$ to $V_\kappa[G]\prec V_\theta[G]$, then one makes the argument work for reflecting cardinals, etc.

Nevertheless, there are a few variant large cardinal notions that are not preserved by small forcing. One large class of such cardinals are the Laver indestructible large cardinals of various types, such as Laver indestructible supercompact cardinals or Laver indestructible measurable cardinals, indestructible weakly compact cardinals and so on. The main results of my papers, Small forcing makes any cardinal superdestructible and Superdestructibility: a dual to Laver indestructibility, show that every Laver indestructible supercompact cardinal is destroyed by small forcing. The general conclusion is that small forcing generally ruins indestructibility.

I suppose that is isn't clear that we should regard the notion of a "Laver indestructible supercompact cardinal" as a large cardinal notion, per se, but I believe that it is sensible to do so. After all, set theorists are often interested in the forcing absoluteness of their concepts, and to make a large cardinal absolute by certain kinds of forcing is a natural strengthening of the concept. But this argument does show that the answer to the question depends on exactly what we count as a large cardinal notion.

Meanwhile, there is also the downward version of the Levy-Solovay theorem, where one asks whether a cardinal can become newly large after small forcing. This also is known in almost all the standard cases, but there remain a few open cases. For example, in my paper Small forcing creates neither strong nor Woodin cardinals, the question remains open whether small forcing can ever increase the degree of strongness of a cardinal (and the only open case is that perhaps a $\lt\lambda$-strong cardinal becomes $\lambda$-strong for a limit $\lambda$ of small cofinality).

The situation with GCH is similar. Almost all the standard large cardinal notions are known to be consistent with the GCH. Indeed, most are provably preserved by the canonical forcing of the GCH. For the smaller large cardinals, one may alternatively appeal to the canonical inner models, which have the large cardinals and the GCH.

Meanwhile, again one may form counterexamples, if one has a more expansive concept of what counts as a large cardinal notion. For example, the existence of a non-measurable weakly measurable cardinal implies $2^\kappa\gt\kappa^+$.

  • $\begingroup$ This was neat, I didn't know about weakly measurables or the work of your student Schanker. $\endgroup$ – Andrés E. Caicedo Jul 22 '13 at 15:03
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    $\begingroup$ Yes, Jason has also introduced the nearly $\theta$-supercomapct cardinals as well. Recently, we proved that the least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact for any desired $\theta$. See jdh.hamkins.org/least-weakly-compact. $\endgroup$ – Joel David Hamkins Jul 22 '13 at 15:11
  • $\begingroup$ Ah, I hadn't seen that one either. Thanks for the link! $\endgroup$ – Andrés E. Caicedo Jul 22 '13 at 15:16