Indestructibility for Ramsey and Ramsey-like cardinals

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V. Gitman and T. A. Johnstone, “Indestructibility properties of Ramsey and Ramsey-like cardinals,” Ann. Pure Appl. Logic, vol. 173, no. 6, pp. Paper No. 103106, 30, 2022.

The study of indestructibility properties of large cardinals was initiated by a seminal result of Lévy and Solovay showing that measurable cardinals cannot be destroyed by small forcing [1]. The Lévy-Solovay phenomenon is now known to extend to most large cardinal notions, which means, in particular, that large cardinals cannot decide ${\rm CH}$ or other independent set theoretic statements that can be manipulated by small forcing. This, taken more generally, is the significance of studying indestructibility properties of large cardinals: it provides a means of verifying which set theoretic properties, among those that can be manipulated by forcing, are compatible with a given large cardinal. There are other applications of indestructibility, such as in separating closely related large cardinal notions by forcing to destroy a part of a large cardinal property, while preserving the rest.

Ramsey cardinals were introduced by Erdős and Hajnal in 1962 [2], who defined that a cardinal $\kappa$ is Ramsey if every coloring $f:[\kappa]^{\lt\omega}\to 2$ of finite tuples of elements of $\kappa$ into two colors has a homogeneous set of size $\kappa$. Ramsey cardinals can also be characterized by the existence of indiscernibles for certain structures as well as by the existence of iterable ultrafilters for certain families of subsets of $\kappa$ of size $\kappa$. Historically very little was known about the indestructibility properties of Ramsey cardinals. A folklore proof, using their original characterization, shows that Ramsey cardinals are indestructible by small forcing [3] (Section 10). Jensen in [4] hinted at a proof that Ramsey cardinals are indestructible by a product forcing which yields the ${\rm GCH}$ in the forcing extension. Finally, Welch showed in [5], using a characterization of Ramsey cardinals in terms of the existence of indiscernibles, that they are indestructible by the forcing to code the universe into a real.

Most general techniques for establishing indestructibility properties of a large cardinal require it to have a characterization in terms of the existence of elementary embeddings. The indestructibility arguments then proceed by showing how to lift (extend) the elementary embedding(s) characterizing the large cardinal from the ground model $V$ to the forcing extension $V[G]$, thus verifying that the large cardinal maintains its property there. It is more common to think of the large cardinals including and above measurable cardinals as being characterized by the existence of elementary embeddings. But in fact, even smaller large cardinals that we typically associate with combinatorial definitions, such as weakly compact and indescribable cardinals, have elementary embedding characterizations. These smaller large cardinals $\kappa$ are usually characterized by the existence of elementary embeddings of weak $\kappa$-models (transitive models of ${\rm ZFC}^-$ of size $\kappa$ with height above $\kappa$) or of $\kappa$-models (additionally closed under $\lt\kappa$-sequences). Mitchell discovered an elementary embeddings characterization of Ramsey cardinals involving the existence of countably complete ultrafilters for weak $\kappa$-models [6], but it was not extensively studied until the first author started to explore it in her dissertation with the purpose of obtaining indestructibility results for Ramsey cardinals [7]. In the process, the first author generalized aspects of the Ramsey embeddings to introduce new Ramsey-like large cardinal notions: $\alpha$-iterable, strongly Ramsey, and super Ramsey cardinals [8]. Since then other Ramsey-like large cardinal notions have been introduced in, for example, [9] and [10].

The elementary embeddings characterization of Ramsey cardinals does not easily lend itself to standard indestructibility techniques. The first difficulty is that the embeddings are on weak $\kappa$-models, as opposed to $\kappa$-models, and these may not even be closed under countable sequences. The second difficulty is that the embeddings are ultrapowers by countably complete ultrafilters and while the lift of an ultrapower embedding remains an ultrapower embedding by a potentially larger ultrafilter, it is not trivial to verify that the larger ultrafilter is still countably complete. Strongly Ramsey and super Ramsey cardinals, which, as the name suggests, are a strengthening of Ramsey cardinals, were defined to have embedding characterizations remedying the deficiencies of Ramsey embeddings with respect to indestructibility arguments. The $\alpha$-iterable cardinals generalized a different aspect of the Ramsey embeddings. They are defined by the existence of partially iterable ultrafilters for weak $\kappa$-models, a requirement that weakens the Ramsey embeddings characterization because countably complete ultrafilters are fully iterable.

In this article, we prove basic indestructibility results for Ramsey, $\alpha$-iterable, strongly Ramsey, and super Ramsey cardinals using a mix of old and newly introduced techniques. We use standard techniques to establish indestructibility properties of strongly Ramsey and super Ramsey cardinals, as their definition was directly motivated to make them easily amenable to these techniques. We develop techniques for lifting embeddings on models without closure. We show that if the forcing notion is countably closed, then a lift of the ultrapower by a countably complete ultrafilter retains this property in the forcing extension. The combination of these new techniques allows us to prove the same basic indestructibility results for Ramsey cardinals as for strongly and super Ramsey cardinals. For the $\alpha$-iterable cardinals, we develop techniques for simultaneously lifting entire iterations of embeddings, so that we can verify that the potentially larger ultrafilter associated with the lift of the first ultrapower in the iteration continues to have at least the iterability of the original ultrafilter. The new indestructibility techniques we introduce can potentially be used to establish a variety of indestructibility results for these and similar large cardinal notions. Here, we obtain the following indestructibility results.

Theorem:

  1. Ramsey, $\alpha$-iterable, strongly Ramsey, and super Ramsey cardinals $\kappa$ are indestructible by:
    1. small forcing,
    2. the canonical forcing of the ${\rm GCH}$,
    3. the forcing to add a fast function on $\kappa$,
  2. If $\kappa$ is one of these large cardinals, then there is a forcing extension in which the large cardinal property of $\kappa$ becomes indestructible by the forcing ${\rm Add}(\kappa,\theta)$ for every cardinal $\theta$.
These indestructibility properties have the following consequences.

Corollary:

  1. If $\kappa$ is Ramsey, $\alpha$-iterable, strongly Ramsey, or super Ramsey, then there is a forcing extension preserving this in which the ${\rm GCH}$ fails at $\kappa$.
  2. If $\kappa$ is Ramsey, $\alpha$-iterable, strongly Ramsey, or super Ramsey, then there is a a forcing extension preserving this in which $\kappa$ is not even weakly compact in ${\rm HOD}$.
  3. If $\kappa$ is Ramsey, then there is a forcing extension destroying this, while preserving that $\kappa$ is virtually Ramsey.
To establish $(3)$, we use techniques from (missing reference). The virtually Ramsey cardinals from $(4)$ were introduced in [11] as an upper bound on the consistency strength of a variant of Chang's Conjecture studied there.

References

  1. A. Lévy and R. M. Solovay, “Measurable cardinals and the continuum hypothesis,” Israel J. Math., vol. 5, pp. 234–248, 1967.
  2. P. Erdős and A. Hajnal, “Some remarks concerning our paper ‘On the structure of set-mappings’. Non-existence of a two-valued $σ$-measure for the first uncountable inaccessible cardinal,” Acta Math. Acad. Sci. Hungar., vol. 13, pp. 223–226, 1962.
  3. A. Kanamori, The higher infinite, Second. Berlin: Springer-Verlag, 2009, p. xxii+536.
  4. R. B. Jensen, “Measurable cardinals and the ${\rm GCH}$,” in Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), Providence, R.I.: Amer. Math. Soc., 1974, pp. 175–178.
  5. A. Beller, R. Jensen, and P. Welch, Coding the universe, vol. 47. Cambridge: Cambridge University Press, 1982, p. i+353.
  6. W. Mitchell, “Ramsey cardinals and constructibility,” J. Symbolic Logic, vol. 44, no. 2, pp. 260–266, 1979.
  7. V. Gitman, Applications of the proper forcing axiom to models of Peano arithmetic. ProQuest LLC, Ann Arbor, MI, 2007, p. 149.
  8. V. Gitman, “Ramsey-like cardinals,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 519–540, 2011.
  9. P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals,” Fund. Math., vol. 242, no. 1, pp. 49–74, 2018.
  10. P. Holy and P. Lücke, “Small Models, Large Cardinals, and Induced Ideals,” Manuscript, 2020.
  11. I. Sharpe and P. Welch, “Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties,” Ann. Pure Appl. Logic, vol. 162, no. 11, pp. 863–902, 2011.