# Blog Archives

# Topic Archive: set theory

# Hierarchies of forcing axioms

I will give an overview over several hierarchies of forcing axioms, with an emphasis on their versions for subcomplete forcing, but in the instances where the concepts are new, their versions for more established classes of forcing, such as proper forcing, are of interest as well. The hierarchies are the traditional one, reaching from the bounded to the unbounded forcing axiom (i.e., versions of Martin’s axiom for classes other than ccc forcing), a hierarchy of resurrection axioms (related to work of Tsaprounis), and (inspired by work of Bagaria, Schindler and Gitman) the “virtual” versions of these hierarchies: the weak bounded forcing axiom hierarchy and the virtual resurrection axiom hierarchy). I will talk about how the levels of these hierarchies are intertwined, in terms of implications or consistency strength. In many cases, I can provide exact consistency strength calculations, which build on techniques to “seal” square sequences, using subcomplete forcing, in the sense that no thread can be added without collapsing ω_{1}. This idea goes back to Todorcevic, in the context of proper forcing (which is completely different from subcomplete forcing).

# Set-theoretic potentialism

In analogy with the ancient views on potential as opposed to actual infinity, set-theoretic potentialism is the philosophical position holding that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-*V _{β}*, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems generally validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true. For example, a Grothendieck-Zermelo universe

*V*, with κ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from

_{κ}*V*just in case κ is a Σ

_{κ}_{3}-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting

*V*.

_{κ}< VThis is current joint work with Øystein Linnebo, in progress, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing. Comments and questions can be made on the speaker’s blog.

# Some necessary applications of logic to operator algebras

Connections between logic and operator algebras in the past century were few and sparse. Recently, some long-standing open problems on the structure of operator algebras were solved using methods from mathematical logic. I will survey some of these results, with a particular emphasis on applications of set theory.

# Cardinal invariants of analytic P-ideals

I will introduce several cardinal invariants associated with analytic P-ideals, concentrating on the ideal of sets of asymptotic density zero. I will give a summary of some recent work on these invariants relating them to the dominating, bounding and splitting numbers, and to some variants of the splitting number. Several fundamental problems remain open and I will try to discuss as many of these as I can.

# Freiling’s axiom of symmetry, or throwing darts at the real line

This will be a talk for the GC Math Program Graduate Student Colloquium. The talk will be aimed at a general audience of mathematics graduate students.

I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if we map every real $x$ to a countable set of reals $A_x$, then there are two reals $x$ and $y$ for which $x$ is not in $A_y$ and $y$ is not in $A_x$. To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis. I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of the negation of CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps from pairs $(x,y)$ to countable sets $A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.

Questions and commentary can be made at: jdh.hamkins.org/freilings-axiom-of-symmetry-graduate-student-colloquium-april-2016/.

# What we talk about when we talk about truth

The goal of set theory, as articulated by Hugh Woodin in his recent Rothschild address at the Isaac Newton Institute, is develop a “convincing philosophy of truth.” There he described the work of set theorists as falling into one of two categories: studying the universe of sets and studying models of set theory. We offer a new perspective on the nature of truth in set theory that may to some extent reconcile these two efforts into one. Joint work with Shoshana Friedman.

# Woodin’s AD-conjecture and local Reinhardt cardinals

We will discuss Woodin’s AD-conjecture, which gives a deep relationship between very large cardinals and determined sets of reals. In particular we will show that the AD-conjecture holds for the axiom I0 and that there are many interesting consequences of this fact. We will also discuss the notion of a local Reinhardt cardinal and how variations of the AD-conjecture might show that such cardinals do not exist.

# Boolean ultrapowers in set theory

Boolean ultrapowers can serve to explain phenomena that arise in the context of iterated ultrapowers, such as the genericity of the critical sequence over the direct limit model. The examples we give are ultrapowers formed using the complete Boolean algebras of Prikry forcing, Magidor forcing, or a generalization of Prikry forcing. Boolean ultrapowers can also be viewed as a direct limit of ultrapowers, and we present some criteria for when the intersection of these ultrapowers is equal to the generic extension of the Boolean ultrapower, thus arriving at a generalization of a phenomenon first observed by Bukovsky and Dehornoy in the context of Prikry forcing.

# Set Theory Day March 11, 2016 at GC

We announce a one-day conference, *Set Theory Day* on Friday, March 11, 2016, in celebration of the 50th birthday of Prof. Joel David Hamkins. The conference will take place at the CUNY Graduate Center and will last all day, with many of Prof. Hamkins’s former students giving talks on their work. All interested mathematicians are invited to attend. The schedule is posted on nylogic.org/set-theory-day. A PDF program is here.

# Ramsey theory and topological dynamics

I will introduce two prominent dynamical systems for a given toplogical group, the greatest ambit and the universal minimal flow, as spaces of (near) ultrafilters on certain Boolean algebras. Representing a topological group as a group of isometries of a highly symmetric structure, I will hint how metrizability and triviality of the universal minimal flow is linked to the (approximate) structural Ramsey property. My focus will lie on problems that arise in the study of universal minimal flows in Ramsey theory, model theory, set theory and continuum theory.

# The continuum hypothesis and other set-theoretic ideas for non-set-theorists

This is a talk for the Einstein Chair Mathematics Seminar at the CUNY Graduate Center, a talk on set theory for non-set-theorists, in two parts:

- An introductory background talk at 11 am
- The main talk at 2 – 4 pm

I shall present several set-theoretic ideas for a non-set-theoretic mathematical audience, focusing particularly on the continuum hypothesis and related issues.

At the morning talk, I shall discuss and prove the Cantor-Bendixson theorem, which asserts that every closed set of reals is the union of a countable set and a perfect set (a closed set with no isolated points), and explain how it led to Cantor’s development of the ordinal numbers and how it establishes that the continuum hypothesis holds for closed sets of reals. We’ll see that there are closed sets of arbitrarily large countable Cantor-Bendixson rank. We’ll talk about the ordinals, about $omega_1$, the long line, and, time permitting, we’ll discuss Suslin’s hypothesis. Dennis has requested that at some point the discussion turn to the role of set theory in the foundation for mathematics, compared for example to that of category theory, and I would look forward to that. I would be prepared also to discuss the Feferman theory in comparison to Grothendieck’s axiom of universes, and other issues relating set theory to category theory.

At the main talk in the afternoon, I’ll begin with a discussion of the continuum hypothesis, including an explanation of the history and logical status of this axiom with respect to the other axioms of set theory, and establish the connection between the continuum hypothesis and Freiling’s axiom of symmetry. I’ll explain the axiom of determinacy and some of its applications and its rich logical situation, connected with large cardinals. I’ll prove the determinacy of open sets and show that AD implies that every set of reals is Lebesgue measurable. I’ll briefly mention the themes and goals of the subjects of cardinal characteristics of the continuum and of Borel equivalence relation theory. If time permits, I’d like to explain some fun geometric decompositions of space that proceed in a transfinite recursion using the axiom of choice, mentioning the open questions concerning whether there can be such decompositions that are Borel.

See also the profile of this talk on my blog.

# Dissertation Defense: Force to change large cardinals

This will be the dissertation defense of the speaker. There will be a one-hour presentation, followed by questions posed by the dissertation committee, and afterwards including some questions posed by the general audience. The dissertation committee consists of Joel David Hamkins (supervisor), Gunter Fuchs, Arthur Apter, Roman Kossak and Philipp Rothmaler.

# Embeddings of the universe into the constructible universe, current state of knowledge

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe. The main question is: can there be an embedding $j:Vto L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $Vneq L$? The notion of *embedding* here is merely that $xin y$ if and only if $j(x)in j(y)$, and such a map need not be elementary nor even $Delta_0$-elementary. It is not difficult to see that there can generally be no $Delta_0$-elementary embedding $j:Vto L$, when $Vneq L$. Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, which shows that every countable model $M$ does admit an embedding $j:Mto L^M$ into its constructible universe. More generally, any two countable models of set theory are comparable; one of them embeds into the other. Indeed, one model $langle M,in^Mrangle$ embeds into another $langle N,in^Nrangle$ just in case the ordinals of the first $text{Ord}^M$ order-embed into the ordinals of the second $text{Ord}^N$. In these theorems, the embeddings $j:Mto L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. Currently, the question remains open, but we have some partial progress, settling it in a number of cases.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut. See more information at the links below:

Blog post for this talk | Related MathOverflow question | Article

# Feedback computability

Many notions of computability allow for an oracle. One natural oracle is the set of algorithms which converge. Normally the way that’s taken is that the programs that are running are different from those in the oracle — the ones running can access an oracle, and the ones in the oracle can’t. But what if the computations running and those in the oracle are the same?