Strongly uplifting cardinals and the boldface resurrection axioms

  • J. D. Hamkins and T. Johnstone, “Strongly uplifting cardinals and the boldface resurrection axioms,” Archive for Mathematical Logic, vol. 56, iss. 7, pp. 1115-1133, 2017.  
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Abstract. We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.

The strongly uplifting cardinals, which we introduce in this article, are a boldface analogue of the uplifting cardinals introduced in our previous paper, Resurrection axioms and uplifting cardinals, and are equivalently characterized as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals. In consistency strength, these new large cardinals lie strictly above the weakly compact, totally indescribable and strongly unfoldable cardinals and strictly below the subtle cardinals, which in turn are weaker in consistency than the existence of $0^\sharp$. The robust diversity of equivalent characterizations of this new large cardinal concept enables constructions and techniques from much larger large cardinal contexts, such as Laver functions and forcing iterations with applications to forcing axioms. Using such methods, we prove that the existence of a strongly uplifting cardinal (or equivalently, a superstrongly unfoldable or almost hugely unfoldable cardinal) is equiconsistent over ZFC with natural instances of the boldface resurrection axioms, including the boldface resurrection axiom for proper forcing, for semi-proper forcing, for c.c.c. forcing and others. Thus, whereas in our prior article we proved that the existence of a mere uplifting cardinal is equiconsistent with natural instances of the (lightface) resurrection axioms, here we adapt both of these notions to the boldface context.

Definitions.

  • An inaccessible cardinal $\kappa$ is strongly uplifting if for every ordinal $\theta$ it is strongly $\theta$-uplifting, which is to say that for every $A\subset V_\kappa$ there is an inaccessible cardinal $\gamma\geq\theta$ and a set $A^*\subset V_\gamma$ such that $\langle V_\kappa,{\in},A\rangle\prec\langle V_\gamma,{\in},A^*\rangle$ is a proper elementary extension.
  • A cardinal $\kappa$ is superstrongly unfoldable, if for every ordinal $\theta$ it is superstrongly $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $V_{j(\kappa)}\subset N$.
  • A cardinal $\kappa$ is almost-hugely unfoldable, if for every ordinal $\theta$ it is almost-hugely $\theta$-unfoldable, which is to say that for each $A\in H_{\kappa^+}$ there is a $\kappa$-model $M$ with $A\in M$ and a transitive set $N$ with an elementary embedding $j:M\to N$ with critical point $\kappa$ and $j(\kappa)\geq\theta$ and $N^{<j(\kappa)}\subset N$.

Remarkably, these different-seeming large cardinal concepts turn out to be exactly equivalent to one another. A cardinal $\kappa$ is strongly uplifting if and only if it is superstrongly unfoldable, if and only if it is almost hugely unfoldable. Furthermore, we prove that the existence of such a cardinal is equiconsistent with several natural instances of the boldface resurrection axiom.

Theorem. The following theories are equiconsistent over ZFC.

  • There is a strongly uplifting cardinal.
  • There is a superstrongly unfoldable cardinal.
  • There is an almost hugely unfoldable cardinal.
  • The boldface resurrection axiom for all forcing.
  • The boldface resurrection axiom for proper forcing.
  • The boldface resurrection axiom for semi-proper forcing.
  • The boldface resurrection axiom for c.c.c. forcing.
  • The weak boldface resurrection axiom for countably-closed forcing, axiom-A forcing, proper forcing and semi-proper forcing, plus $\neg\text{CH}$.

 

 

Resurrection axioms and uplifting cardinals

  • J. D. Hamkins and T. Johnstone, “Resurrection axioms and uplifting cardinals,” Archive for Mathematical Logic, vol. 53, iss. 3-4, p. p.~463–485, 2014.  
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Abstract. We introduce the resurrection axioms, a new class of forcing axioms, and the uplifting cardinals, a new large cardinal notion, and prove that various instances of the resurrection axioms are equiconsistent over ZFC with the existence of uplifting cardinal.

Many classical forcing axioms can be viewed, at least informally, as the claim that the universe is existentially closed in its forcing extensions, for the axioms generally assert that certain kinds of filters, which could exist in a forcing extension $V[G]$, exist already in $V$. In several instances this informal perspective is realized more formally: Martin’s axiom is equivalent to the assertion that $H_{\frak{c}}$ is existentially closed in all c.c.c. forcing extensions of the universe, meaning that $H_{\frak{c}}\prec_{\Sigma_1}V[G]$ for all such extensions; the bounded proper forcing axiom is equivalent to the assertion that $H_{\omega_2}$ is existentially closed in all proper forcing extensions, or $H_{\omega_2}\prec_{\Sigma_1}V[G]$; and there are other similar instances.

In model theory, a submodel $M\subset N$ is existentially closed in $N$ if existential assertions true in $N$ about parameters in $M$ are true already in $M$, that is, if $M$ is a $\Sigma_1$-elementary substructure of $N$, which we write as $M\prec_{\Sigma_1} N$. Furthermore, in a general model-theoretic setting, existential closure is tightly connected with resurrection, the theme of this article.

Elementary Fact. If $\mathcal{M}$ is a submodel of $\mathcal{N}$, then the following are equivalent.

  1. The model $\mathcal{M}$ is existentially closed in $\mathcal{N}$.
  2. $\mathcal{M}\subset \mathcal{N}$ has resurrection. That is, there is a further extension $\mathcal{M}\subset\mathcal{N}\subset\mathcal{M}^+$ for which $\mathcal{M}\prec\mathcal{M}^+$.

We call this resurrection because although certain truths in $\mathcal{M}$ may no longer hold in the extension $\mathcal{N}$, these truths are nevertheless revived in light of $\mathcal{M}\prec\mathcal{M}^+$ in the further extension to $\mathcal{M}^+$.

In the context of forcing axioms, we are more interested in the case of forcing extensions than in the kind of arbitrary extension $\mathcal{M}^+$ arising in the fact, and in this context the equivalence of (1) and (2) breaks own, although the converse implication $(2)\to(1)$ always holds, and every instance of resurrection implies the corresponding instance of existential closure. This key observation leads us to the main unifying theme of this article, the idea that

resurrection may allow us to formulate more robust forcing axioms 

than existential closure or than combinatorial assertions about filters and dense sets. We therefore introduce in this paper a spectrum of new forcing axioms utilizing the resurrection concept.

Main Definition. Let $\Gamma$ be a fixed definable class of forcing notions.

  1. The resurrection axiom $\text{RA}(\Gamma)$ is the assertion that for every forcing notion $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, with $\vdash_{\mathbb{Q}}\mathbb{R}\in\Gamma$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.
  2. The weak resurrection axiom $\text{wRA}(\Gamma)$ is the assertion that for every $\mathbb{Q}\in\Gamma$ there is further forcing $\mathbb{R}$, such that if $g\ast h\subset\mathbb{Q}\ast\mathbb{R}$ is $V$-generic, then $H_{\frak{c}}\prec H_{\frak{c}}^{V[g\ast h]}$.

The main result is to prove that various formulations of the resurrection axioms are equiconsistent with the existence of an uplifting cardinal, where an inaccessible cardinal $\kappa$ is uplifting, if there are arbitrarily large inaccessible cardinals $\gamma$ for which $H_\kappa\prec H_\gamma$.  This is a rather weak large cardinal notion, having consistency strength strictly less than the existence of a Mahlo cardinal, which is traditionally considered to be very low in the large cardinal hierarchy.  One highlight of the article is our development of “the world’s smallest Laver function,” the Laver function concept for uplifting cardinals, and we perform an analogue of the Laver preparation in order to achieve the resurrection axiom for c.c.c. forcing.

Main Theorem. The following theories are equiconsistent over ZFC:

  1. There is an uplifting cardinal.
  2. $\text{RA}(\text{all})$.
  3. $\text{RA}(\text{ccc})$.
  4. $\text{RA}(\text{semiproper})+\neg\text{CH}$.
  5. $\text{RA}(\text{proper})+\neg\text{CH}$.
  6. For some countable ordinal $\alpha$, the axiom $\text{RA}(\alpha\text{-proper})+\neg\text{CH}$.
  7. $\text{RA}(\text{axiom-A})+\neg\text{CH}$.
  8. $\text{wRA}(\text{semiproper})+\neg\text{CH}$.
  9. $\text{wRA}(\text{proper})+\neg\text{CH}$.
  10. For some countable ordinal $\alpha$, the axiom $\text{wRA}(\alpha\text{-proper})+\neg\text{CH}$.
  11. $\text{wRA}(\text{axiom-A})+\neg\text{CH}$.
  12. $\text{wRA}(\text{countably closed})+\neg\text{CH}$.

The proof outline proceeds in two directions: on the one hand, the resurrection axioms generally imply that the continuum $\frak{c}$ is uplifting in $L$; and conversely, given any uplifting cardinal $\kappa$, we may perform a suitable lottery iteration of $\Gamma$ forcing to obtain the resurrection axiom for $\Gamma$ in a forcing extension with $\kappa=\frak{c}$.

In a follow-up article, currently nearing completion, we treat the boldface resurrection axioms, which allow a predicate $A\subset\frak{c}$ and ask for extensions of the form $\langle H_{\frak{c}},{\in},A\rangle\prec\langle H_{\frak{c}}^{V[g\ast h]},{\in},A^\ast\rangle$, for some $A^\ast\subset\frak{c}^{V[g\ast h]}$ in the extension.  In that article, we prove the equiconsistency of various formulations of boldface resurrection with the existence of a strongly uplifting cardinal, which we prove is the same as a superstrongly unfoldable cardinal.

Thomas Johnstone

Thomas Johnstone earned his Ph.D. under my supervision in June, 2007 at the CUNY Graduate Center.  Tom likes to get thoroughly to the bottom of a problem, and this indeed is what he did in his dissertation work on the forcing-theoretic aspects of unfoldable cardinals.  He seemed to want always to dig deeper, seeking out the unstated general phenomenon behind the results.  His characteristic style of giving a seminar talk—pure mathematical pleasure to attend—is to explain not only why the mathematical fact is true, but also why the proof must be the way that it is.  Thomas holds a tenure-track position at the New York City College of Technology of CUNY.

Thomas A. Johnstone

web page | math genealogy | MathSciNet | ar$\chi$iv | google scholar | related posts
 

Thomas A. Johnstone, “Strongly unfoldable cardinals made indestructible,” Ph.D. dissertation, The Graduate Center of the City University of New York, June 2007.

Abstract. I provide indestructibility results for weakly compact, indescribable and strongly unfoldable cardinals. In order to make these large cardinals indestructible, I assume the existence of a strongly unfoldable cardinal $\kappa$, which is a hypothesis consistent with $V=L$. The main result shows that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${<}\kappa$-closed forcing which does not collapse $\kappa^{+}$. As strongly unfoldable cardinals strengthen both indescribable and weakly compact cardinals, I obtain indestructibility for these cardinals also, thereby reducing the large cardinal hypothesis of previously known indestructibility results for these cardinals significantly. Finally, I use the developed methods to show the consistency of a weakening of the Proper Forcing Axiom $\rm PFA$ relative to the existence of a strongly unfoldable cardinal.

What is the theory ZFC without power set?

  • V. Gitman, J. D. Hamkins, and T. A.~Johnstone, “What is the theory ZFC without Powerset?,” Mathematical Logic Quarterly, vol. 62, iss. 4–5, pp. 391-406, 2016.  
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This is joint work with Victoria Gitman and Thomas Johnstone.

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC- in which $\omega_1$ is singular, in which every set of reals is countable, yet $\omega_1$ exists, in which there are sets of reals of every size $\aleph_n$, but none of size $\aleph_\omega$, and therefore, in which the collection axiom sceme fails; there are models of ZFC- for which the Los theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC- for which the Gaifman theorem fails, in that there is an embedding $j:M\to N$ of ZFC- models that is $\Sigma_1$-elementary and cofinal, but not elementary; there are elementary embeddings $j:M\to N$ of ZFC- models whose cofinal restriction $j:M\to \bigcup j“M$ is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC- to a $\Sigma_1$-formula or a $\Pi_1$-formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory $\text{ZFC}^-$, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.

See Victoria Gitman’s summary post on the article

Indestructible strong unfoldability

  • J. D. Hamkins and T. A. Johnstone, “Indestructible strong unfoldability,” Notre Dame J.~Form.~Log., vol. 51, iss. 3, pp. 291-321, 2010.  
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Using the lottery preparation, we prove that any strongly unfoldable cardinal $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed + $\kappa^+$-preserving forcing. This degree of indestructibility, we prove, is the best possible from this hypothesis within the class of ${\lt}\kappa$-closed forcing. From a stronger hypothesis, however, we prove that the strong unfoldability of $\kappa$ can be made indestructible by all ${\lt}\kappa$-closed forcing. Such indestructibility, we prove, does not follow from indestructibility merely by ${\lt}\kappa$-directed closed forcing. Finally, we obtain global and universal forms of indestructibility for strong unfoldability, finding the exact consistency strength of universal indestructibility for strong unfoldability.

The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$

  • J. D. Hamkins and T. A. Johnstone, “The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$,” Proc.~Amer.~Math.~Soc., vol. 137, iss. 5, pp. 1823-1833, 2009.  
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We prove that the PFA lottery preparation of a strongly unfoldable cardinal $\kappa$ under $\neg 0^\sharp$ forces $\text{PFA}(\aleph_2\text{-preserving})$, $\text{PFA}(\aleph_3\text{-preserving})$ and $\text{PFA}_{\aleph_2}$, with $2^\omega=\kappa=\aleph_2$.  The method adapts to semi-proper forcing, giving $\text{SPFA}(\aleph_2\text{-preserving})$, $\text{SPFA}(\aleph_3\text{-preserving})$ and $\text{SPFA}_{\aleph_2}$ from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent with the conjunction $\text{SPFA}(\aleph_2\text{-preserving})+\text{SPFA}(\aleph_3\text{-preserving})+\text{SPFA}_{\aleph_2}+2^\omega=\aleph_2$.  Since unfoldable cardinals are relatively weak as large cardinal notions, our summary conclusion is that in order to extract significant strength from PFA or SPFA, one must collapse $\aleph_3$ to $\aleph_1$.

What is the theory of ZFC-Powerset? Toronto 2011

This was a talk at the Toronto Set Theory Seminar held April 22, 2011 at the Fields Institute in Toronto.

The theory ZFC-, consisting of the usual axioms of ZFC but with the powerset axiom removed, when axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the axiom of choice, is weaker than commonly supposed, and suffices to prove neither that a countable union of countable sets is countable, nor that $\omega_1$ is regular, nor that the Los theorem holds for ultrapowers, even for well-founded ultrapowers on a measurable cardinal, nor that the Gaifman theorem holds, that is, that every $\Sigma_1$-elementary cofinal embedding $j:M\to N$ between models of the theory is fully elementary, nor that $\Sigma_n$ sets are closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using the collection axiom rather than replacement in the axiomatization above. These results extend prior work of Zarach. This is joint work with Victoria Gitman and Thomas Johnstone.

Article | Victoria Gitman’s post