# Kaethe Lynn Bruesselbach Minden, PhD 2017, CUNY Graduate Center

Kaethe Lynn Bruesselbach Minden successfully defended her dissertation on April 7, 2017 at the CUNY Graduate Center, under the supervision of Professor Gunter Fuchs. I was a member of the dissertation committee, along with Arthur Apter.

Her defense was impressive!  She was a master of the entire research area, ready at hand with the technical details to support her account of any topic that arose.

Kaethe Minden, “On Subcomplete Forcing,” Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2017. (arxiv/1705.00386)

Abstract. I survey an array of topics in set theory and their interaction with, or in the context of, a novel class of forcing notions: subcomplete forcing. Subcomplete forcing notions satisfy some desirable qualities; for example they don’t add any new reals to the model, and they admit an iteration theorem. While it is straightforward to show that any forcing notion which is countably closed is also subcomplete, it turns out that other well-known, more subtle forcing notions like Prikry forcing and Namba forcing are also subcomplete. Subcompleteness was originally defined by Ronald Björn Jensen around 2009. Jensen’s writings make up the vast majority of the literature on the subject. Indeed, the definition in and of itself is daunting. I have attempted to make the subject more approachable to set theorists, while showing various properties of subcomplete forcing which one might desire of a forcing class.

It is well-known that countably closed forcings cannot add branches through $\omega_1$-trees. I look at the interaction between subcomplete forcing and $\omega_1$-trees. It turns out that sub-complete forcing also does not add cofinal branches to $\omega_1$-trees. I show that a myriad of other properties of trees of height $\omega_1$ as explored in [FH09] are preserved by subcomplete forcing; for example, I show that the unique branch property of Suslin trees is preserved by subcomplete forcing.

Another topic I explored is the Maximality Principle ($\text{MP}$). Following in the footsteps of Hamkins [Ham03], Leibman [Lei], and Fuchs [Fuc08], [Fuc09], I examine the subcomplete maximality principle. In order to elucidate the ways in which subcomplete forcing generalizes the notion of countably closed forcing, I compare the countably closed maximality principle ($\text{MP}_{<\omega_1\text{-closed}}$) to the subcomplete maximality principle ($\text{MP}_{sc}$). Again, since countably closed forcing is subcomplete, this is a natural question to ask. I was able to show that many of the results about $\text{MP}_{<\omega_1\text{-closed}}$ also hold for $\text{MP}_{sc}$; for example, the boldface appropriate notion of $\text{MP}_{sc}$ is equiconsistent with a fully reflecting cardinal. However, it is not the case that there are direct implications between the subcomplete and countably closed maximality principles.

Another forcing principle explored in my thesis is the Resurrection Axiom ($\text{RA}$). Hamkins and Johnstone [HJ14a] defined the resurrection axiom only relative to $H_{\mathfrak{c}}$, and focus mainly on the resurrection axiom for proper forcing. They also show the equiconsistency of various resurrection axioms with an uplifting cardinal. I argue that the subcomplete resurrection axiom should naturally be considered relative to $H_{\omega_2}$, and showed that the subcomplete resurrection axiom is equiconsistent with an uplifting cardinal.

A question reasonable to ask about any class of forcings is whether or not the resurrection axiom and the maximality principle can consistently both hold for that class. I originally had this question about the full principles, not restricted to any class, but in my thesis it was appropriate to look at the question for subcomplete forcing. I answer the question positively for subcomplete forcing using a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals needed to force the principles separately. I show that the boldface versions of $\text{MP}_{sc}+\text{RA}_{sc}$ both holding is equiconsistent with the existence of a strongly uplifting fully reflecting cardinal. While Jensen [Jen14] shows that Prikry forcing is subcomplete, I long suspected that many variants of Prikry forcing which have a kind of genericity criterion are also subcomplete. After much work I managed to show that a variant of Prikry forcing known as Diagonal Prikry Forcing is subcomplete, giving another example of subcomplete forcing to add to the list.

Kaethe has taken up a faculty position at Marlboro College in Vermont.

# Miha E. Habič, PhD 2017, CUNY Graduate Center

Miha E. Habič successfully defended his dissertation under my supervision at the CUNY Graduate Center on April 7th, 2017, earning his Ph.D. degree in May 2017.

It was truly a pleasure to work with Miha, who is an outstanding young mathematician with enormous promise. I shall look forward to seeing his continuing work.

Miha E. Habič, “Joint Laver diamonds and grounded forcing axioms,”  Ph.D. dissertation for The Graduate Center of the City University of New York, May, 2017 (arxiv:1705.04422).

Abstract. In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for $\kappa$ is joint if for any sequence of targets there is a single elementary embedding $j$ with critical point $\kappa$ such that each Laver diamond guesses its respective target via $j$. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for $\kappa$ yields a joint sequence of length $\kappa$, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of $\theta$-strong cardinals where, for certain $\theta$, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary $\diamondsuit_\kappa$-sequences on any regular cardinal $\kappa$. The main result concerning these shows that there is no separation according to length and a single $\diamondsuit_\kappa$-sequence yields joint families of all possible lengths.

In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin’s axiom. This grounded Martin’s axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin’s axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin’s axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin’s axiom itself.

Miha will shortly begin a post-doctoral research position at Charles University in Prague.

# Giorgio Audrito, PhD 2016, University of Torino

Dr. Giorgio Audrito has successfully defended his dissertation, “Generic large cardinals and absoluteness,” at the University of Torino under the supervision of Matteo Viale.

The dissertation Examing Board consisted of myself (serving as Presidente), Alessandro Andretta and Sean Cox.  The defense took place March 2, 2016.

The dissertation was impressive, introducing (in joint work with Matteo Viale) the iterated resurrection axioms $\text{RA}_\alpha(\Gamma)$ for a forcing class $\Gamma$, which extend the idea of the resurrection axioms from my work with Thomas Johnstone, The resurrection axioms and uplifting cardinals, by making successive extensions of the same type, forming the resurrection game, and insisting that that the resurrection player have a winning strategy with game value $\alpha$. A similar iterative game idea underlies the $(\alpha)$-uplifting cardinals, from which the consistency of the iterated resurrection axioms can be proved. A final chapter of the dissertation (joint with Silvia Steila), develops the notion of $C$-systems of filters, generalizing the more familiar concepts of extenders and towers.

# 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.
@ARTICLE{HamkinsJohnstone2017:StronglyUpliftingCardinalsAndBoldfaceResurrection,
author = {Joel David Hamkins and Thomas Johnstone},
title = {Strongly uplifting cardinals and the boldface resurrection axioms},
journal="Archive for Mathematical Logic",
year="2017",
month="Nov",
day="01",
volume="56",
number="7",
pages="1115--1133",
eprint = {1403.2788},
archivePrefix = {arXiv},
primaryClass = {math.LO},
issn="1432-0665",
doi="10.1007/s00153-017-0542-y",
url = {http://wp.me/p5M0LV-IE},
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.",
keywords = {},
source = {},
}

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.
@ARTICLE{HamkinsJohnstone2014:ResurrectionAxiomsAndUpliftingCardinals,
AUTHOR = "Joel David Hamkins and Thomas Johnstone",
TITLE = "Resurrection axioms and uplifting cardinals",
JOURNAL = "Archive for Mathematical Logic",
publisher= {Springer},
YEAR = "2014",
volume = "53",
number = "3-4",
pages = "p.~463--485",
month = "",
note = "",
url = "http://jdh.hamkins.org/resurrection-axioms-and-uplifting-cardinals",
eprint = "1307.3602",
archivePrefix = {arXiv},
primaryClass = {math.LO},
doi= "10.1007/s00153-014-0374-y",
issn= {0933-5846},
abstract = "",
keywords = "",
source = "",
file = F,
}

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.

# Recent progress on the modal logic of forcing and grounds, CUNY Logic Workshop, September 2012

This will be a talk for the CUNY Logic Workshop on September 7, 2012.

Abstract. The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such asthe assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, orthe assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC.  Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”.  In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.

# Moving up and down in the generic multiverse

• J. D. Hamkins and B. Löwe, “Moving up and down in the generic multiverse,” Logic and its Applications, ICLA 2013 LNCS, vol. 7750, pp. 139-147, 2013.
@ARTICLE{HamkinsLoewe2013:MovingUpAndDownInTheGenericMultiverse,
AUTHOR = {Joel David Hamkins and Benedikt Löwe},
title = {Moving up and down in the generic multiverse},
journal = {Logic and its Applications, ICLA 2013 LNCS},
publisher= {Springer},
editor= {Lodaya, Kamal},
isbn= {978-3-642-36038-1},
year = {2013},
volume = {7750},
number = {},
pages = {139--147},
doi= {10.1007/978-3-642-36039-8_13},
month = {},
note = {},
url = {http://wp.me/p5M0LV-od},
eprint = {1208.5061},
archivePrefix = {arXiv},
primaryClass = {math.LO},
abstract = {},
keywords = {},
source = {},
}

In this extended abstract we investigate the modal logic of the generic multiverse, which is a bimodal logic with operators corresponding to the relations “is a forcing extension of”‘ and “is a ground model of”. The fragment of the first relation is the modal logic of forcing and was studied by us in earlier work. The fragment of the second relation is the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.

The main theorems are as follows:

Theorem.  If  ZFC is consistent, then there is a model of  ZFC  whose modal logic of forcing and modal logic of grounds are both S4.2.

Theorem.  If  the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S4.2 and whose modal logic of grounds is S5.

Theorem.  If  the theory “$L_\delta\prec L+\delta$ is inaccessible” is consistent, then there is a model of set theory whose modal logic of forcing is S5 and whose modal logic of grounds is S4.2.

Theorem. There is no model of set theory such that both its modal logic of forcing and its modal logic of grounds are S5.

The current article is a brief extended abstract (10 pages).  A fuller account with more detailed proofs and further information will be provided in a subsequent articl

eprints:  ar$\chi$iv | NI12059-SAS | Hamburg #450

# Victoria Gitman

Victoria Gitman earned her Ph.D. under my supervision at the CUNY Graduate Center in June, 2007.  For her dissertation work, Victoria had chosen a very difficult problem, the 1962 question of Dana Scott to characterize the standard systems of models of Peano Arithmetic, a question in the field of models of arithmetic that had been open for over forty years. Victoria was able to make progress, now published in several papers, by using an inter-disciplinary approach, applying set-theoretic ideas—including a use of the proper forcing axiom PFA—to the problem in the area of models of arithmetic, where such methods hadn’t often yet arisen.  Ultimately, she showed under PFA that every arithmetically closed proper Scott set is the standard system of a model of PA.  This result extends the classical result to a large new family of Scott sets, providing for these sets an affirmative solution to Scott’s problem.  In other dissertation work, Victoria untangled the confusing mass of ideas surrounding various Ramsey-like large cardinal concepts, ultimately separating them into a beautiful hierarchy, a neighborhood of the vast large cardinal hierarchy intensely studied by set theorists.  (Please see the diagram in her dissertation.)  Victoria holds a tenure-track position at the New York City College of Technology of CUNY.

Victoria Gitman

Victoria Gitman, “Applications of the Proper Forcing Axiom to Models of Peano Arithmetic,”  Ph.D. dissertation for the Graduate Center of the City University of New York, June 2007.

Abstract. In Chapter 1, new results are presented on Scott’s Problem in the subject of models of Peano Arithmetic. Some forty years ago, Dana Scott showed that countable Scott sets are exactly the countable standard systems of models of PA, and two decades later, Knight and Nadel extended his result to Scott sets of size $\omega_1$. Here it is shown that assuming the Proper Forcing Axiom, every arithmetically closed proper Scott set is the standard system of a model of PA. In Chapter 2, new large cardinal axioms, based on Ramsey-like embedding properties, are introduced and placed within the large cardinal hierarchy. These notions generalize the seldom encountered embedding characterization of Ramsey cardinals. I also show how these large cardinals can be used to obtain indestructibility results for Ramsey cardinals.

# Jonas Reitz

Jonas Reitz earned his Ph.D under my supervision in June, 2006 at the CUNY Graduate Center.  He was truly a pleasure to supervise. From the earliest days of his dissertation research, he had his own plan for the topic of the work: he wanted to “undo” forcing, to somehow force backwards, from the extension to the ground model. At first I was skeptical, but in time, ideas crystalized around the ground axiom (now with its own Wikipedia entry), formulated using a recent-at-the-time result of Richard Laver.  Along with Laver’s theorem, Jonas’s dissertation was the beginning of the body of work now known as set-theoretic geology.  Jonas holds a tenured position at the New York City College of Technology of CUNY.

Jonas Reitz

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

Jonas Reitz, “The ground axiom,” Ph.D. dissertation, CUNY Graduate Center, June, 2006.  ar$\chi$iv

Abstract.  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.

# George Leibman

George Joseph Leibman earned his Ph.D. under my supervision in June, 2004 at the CUNY Graduate Center. He was my first Ph.D. student. Being very interested both in forcing and in modal logic, it was natural for him to throw himself into the emerging developments at the common boundary of these topics.  He worked specifically on the natural extensions of the maximality principle where when one considers a fixed definable class $\Gamma$ of forcing notions.  This research engaged with fundamental questions about the connection between the forcing-theoretic properties of the forcing class $\Gamma$ and the modal logic of its forcing validities, and was a precursor of later work, including joint work, on the modal logic of forcing.

George Leibman

George Leibman, “Consistency Strengths of Modified Maximality Principles,” Ph.D. thesis, CUNY Graduate Center, 2004.  ar$\chi$iv

Abstract. The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension $V^{\mathbb{P}}$, and remains true in any further forcing extension of $V^{\mathbb{P}}$, then it is true in all forcing extensions of $V$.  A modified maximality principle $\text{MP}_\Gamma$ arises when considering forcing with a particular class $\Gamma$ of forcing notions. A parametrized form of such a principle, $\text{MP}_\Gamma(X)$, considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set $X$ which depends on the forcing class $\Gamma$ being considered. A stronger necessary form of such a principle, $\square\text{MP}_\Gamma(X)$, occurs when it continues to be true in all $\Gamma$ forcing extensions.

This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, COHEN, COLL (the forcing notions that collapse ordinals to $\omega$), ${\lt}\kappa$ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered.

# 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.
@ARTICLE{HamkinsJohnstone2009:PFA(aleph_2-preserving),
AUTHOR = {Hamkins, Joel David and Johnstone, Thomas A.},
TITLE = {The proper and semi-proper forcing axioms for forcing notions that preserve {$\aleph_2$} or {$\aleph_3$}},
JOURNAL = {Proc.~Amer.~Math.~Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {137},
YEAR = {2009},
NUMBER = {5},
PAGES = {1823--1833},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E55 (03E40)},
MRNUMBER = {2470843 (2009k:03087)},
MRREVIEWER = {John Krueger},
DOI = {10.1090/S0002-9939-08-09727-X},
URL = {http://wp.me/p5M0LV-3v},
file = F,
}

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 $\aleph_1$.

# The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal

• J. D. Hamkins and W. Woodin, “The necessary maximality principle for c.c.c.~forcing is equiconsistent with a weakly compact cardinal,” Math.~Logic Q., vol. 51, iss. 5, pp. 493-498, 2005.
@ARTICLE{HamkinsWoodin2005:NMPccc,
AUTHOR = {Joel David Hamkins and W.~Hugh Woodin},
TITLE = {The necessary maximality principle for c.c.c.~forcing is equiconsistent with a weakly compact cardinal},
JOURNAL = {Math.~Logic Q.},
FJOURNAL = {Mathematical Logic Quarterly},
VOLUME = {51},
YEAR = {2005},
NUMBER = {5},
PAGES = {493--498},
ISSN = {0942-5616},
MRCLASS = {03E65 (03E55)},
MRNUMBER = {2163760 (2006f:03082)},
MRREVIEWER = {Tetsuya Ishiu},
DOI = {10.1002/malq.200410045},
URL = {http://wp.me/s5M0LV-nmpccc},
eprint = {math/0403165},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.

See related article on the Maximality Principle

# A simple maximality principle

• J. D. Hamkins, “A simple maximality principle,” Journal of Symbolic Logic, vol. 68, iss. 2, pp. 527-550, 2003.
@article{Hamkins2003:MaximalityPrinciple,
AUTHOR = {Hamkins, Joel David},
TITLE = {A simple maximality principle},
JOURNAL = {Journal of Symbolic Logic},
VOLUME = {68},
YEAR = {2003},
NUMBER = {2},
PAGES = {527--550},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E35 (03E40)},
MRNUMBER = {1976589 (2005a:03094)},
MRREVIEWER = {Ralf-Dieter Schindler},
DOI = {10.2178/jsl/1052669062},
URL = {http://wp.me/p5M0LV-2v},
month = {},
eprint = {math/0009240},
archivePrefix = {arXiv},
primaryClass = {math.LO},
}

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence$\varphi$ holding in some forcing extension $V^{\mathbb{P}}$ and all subsequent extensions $V^{\mathbb{P}*\mathbb{Q}}$ holds already in $V$. It follows, in fact, that such sentences must also hold in all forcing extensions of $V$. In modal terms, therefore, the Maximality Principle is expressed by the scheme $(\Diamond\Box\varphi)\to\Box\varphi$, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in $\varphi$, is equiconsistent with the scheme asserting that $V_\delta$ is an elementary substructure of $V$ for an inaccessible cardinal $\delta$, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that $0^\sharp$ exists, that $x^\sharp$ exists for every set $x$, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.