# Blog Archives

# Topic Archive: forcing

# Functors and infinitary interpretations of structures

It has long been recognized that the existence of an interpretation of one countable structure B in another one A yields a homomorphism from the automorphism group Aut(A) into Aut(B). Indeed, it yields a functor from the category Iso(A) of all isomorphic copies of A (under isomorphisms) into the category Iso(B). In traditional model theory, the converse is false. However, when we extend the concept of interpretation to allow interpretations by L_{ω1ω} formulas, we find that now the converse essentially holds: every Borel functor arises from an infinitary interpretation of B in A, and likewise every Borel-measurable homomorphism from Aut(A) into Aut(B) arises from such an interpretation. Moreover, the complexity of an interpretation matches the complexities of the corresponding functor and homomorphism. We will discuss the concepts and the forcing necessary to prove these results and other corollaries.

# A countable ordinal definable set of reals without ordinal definable elements

In 2010, a question on MathOverflow asked whether it is possible that a countable ordinal definable set of reals has elements that are not ordinal definable. It is easy to see that every element of a finite ordinal definable set of reals is itself ordinal definable. Also it is consistent that there is an uncountable ordinal definable set of reals without ordinal definable elements. It turned out that the question for countable sets of reals was not known. It was finally solved by Kanovei and Lyubetsky in 2014, who showed, using a forcing extension by a finite-support product of Jensen reals, that it is consistent to have a countable ordinal definable set of reals without ordinal definable elements. In the talk, I will give full details of their argument.

An extended abstract is available on my blog here.

# Characterizing forcing extensions

I shall present a proof of a theorem of Bukovský from 1973 that characterizes the set-forcing extensions among all pairs of ZFC models $M\subseteq N$: these are precisely the pairs satisfying a uniform covering property. His result has recently resurfaced in the study of set-theoretic geology and can, for example, also be used to give a conceptual proof of (a version of) the intermediate model theorem.

# $V$ need not be a forcing extension of $\mathrm{HOD}$ or of the mantle

In 1972 Vopenka showed that $V$ is a union of set-generic extensions of $\mathrm{HOD}$ by establishing that every set in $V\setminus\mathrm{HOD}$ is set generic over $\mathrm{HOD}$. It is natural to consider whether that union can be replaced by a single forcing, possibly a proper class, over $\mathrm{HOD}$. In 2012 Friedman showed that $V$ is a class forcing extension of $\mathrm{HOD}$ by a partial order definable in $V$ – however, this leaves open the question of whether such a partial order can be defined in $\mathrm{HOD}$ itself. In this talk I will show that the qualifier ‘in $V$’ is necessary in Friedman’s theorem, by producing a model which is not class generic over $\mathrm{HOD}$ for any forcing definable in $\mathrm{HOD}$.

In the area of set theory known as set-theoretic geology, the mantle $M$ (the intersection of all grounds) is an inner model that enjoys a relationship to $V$ similar to that of $\mathrm{HOD}$, but ‘in the opposite direction’ – every set not in $M$ is omitted by a ground of $V$. Does it follow that we can build $V$ up over $M$ by iteratively adding those sets back in via forcing? In particular, does it follow that $V$ is a class forcing extension of $M$? The example produced in this talk will show that the answer is no – there is a model of set theory $V$ which is not a class forcing extension of $M$ by any forcing definable in $M$.

# 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 geology and the downward-directed grounds hypothesis: part II

I will continue presenting Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. See the main abstract at Set-theoretic geology and the downward directed ground hypothesis.

See my blog post about this talk.

# Set-theoretic geology and the downward-directed grounds hypothesis

Forcing is often viewed as a method of constructing larger models extending a given model of set theory. The topic of set-theoretic geology inverts this perspective by investigating how the current set-theoretic universe $V$ might itself have arisen as a forcing extension of an inner model. Thus, an inner model $W\subset V$ is a ground of $V$ if we can realize $V=W[G]$ as a forcing extension of $W$ by some $W$-generic filter $G\subset\mathbb Q\in W$. Reitz had inquired in his dissertation whether any two grounds of $V$ must have a common deeper ground. Fuchs, myself and Reitz introduced the downward-directed grounds hypothesis, which asserts a positive answer, even for any set-indexed collection of grounds, and we showed that this axiom has many interesting consequences for set-theoretic geology.

I shall give a complete detailed account of Toshimichi Usuba’s recent proof of the strong downward-directed grounds hypothesis. This breakthrough result answers what had been for ten years the central open question in the area of set-theoretic geology and leads immediately to numerous consequences that settle many other open questions in the area, as well as to a sharpening of some of the central concepts of set-theoretic geology, such as the fact that the mantle coincides with the generic mantle and is a model of ZFC. I shall also present Usuba’s related result that if there is a hyper-huge cardinal, then there is a bedrock model, a smallest ground. I find this to be a surprising and incredible result, as it shows that large cardinal existence axioms have consequences on the structure of grounds for the universe.

See my blog post about this talk.

# Virtual large cardinals

Given a very large cardinal property $\mathcal A$, e.g. supercompact or extendible, characterized by the existence of suitable set-sized embeddings, we define that a cardinal $\kappa$ is *virtually* $\mathcal A$ if the embeddings characterizing $\mathcal A$ exist in some set-forcing extension. In this terminology, the remarkable cardinals introduced by Schindler, which he showed to be equiconsistent with the absoluteness of the theory of $L(\mathbb R)$ under proper forcing, are virtually supercompact. We introduce the notions of *virtually extendible*, *virtually $n$-huge*, and *virtually rank-into-rank* cardinals and study their properties. In the realm of virtual large cardinals, we can even go beyond the Kunen Inconsistency because it is possible that in a set-forcing extension there is an embedding $j:V_\delta^V\to V_\delta^V$ with $\delta>\lambda+1$, where $\lambda$ is the supremum of the critical sequence. The virtual large cardinals are much smaller than their (possibly inconsistent) counterparts. Silver indiscernibles possess all the virtual large cardinal properties we will consider, and indeed the large cardinals are downward absolute to $L$. We give a tight measure on the consistency strength of the virtual large cardinals in terms of the $\alpha$-iterable cardinals hierarchy. Virtual large cardinals can be used, for instance, to measure the consistency strength of the *Generic Vopěnka’s Principle*, introduced by Bagaria, Schindler, and myself, which states that for every proper class $\mathcal C$ of structures of the same type, there are $B\neq A$ both in $\mathcal C$ such that $B$ embeds into $A$ in some set-forcing extension. This is joint work with Ralf Schindler.

# Killing measurable and supercompact cardinals softly

This talk follows the theme of killing-them-softly between set-theoretic universes. The main theorems in this theme show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree, for a variety of large cardinals including inaccessible, Mahlo, measurable and supercompact. The killing-them-softly theme is about both forcing and the gradations in large cardinal strength. This talk will focus on measurable and supercompact cardinals, and follows the larger theme of exploring interactions between large cardinals and forcing which is central to modern set theory.

# Singular in V, regular and non-measurable in HOD

Getting a model where $\kappa$ is singular in $V$ but measurable in ${\rm HOD}$ is somewhat straightforward however ensuring that $\kappa$ is regular but not measurable in ${\rm HOD}$ is a surprisingly more difficult problem. Magidor navigated around the issues and I will present his result starting with one measurable. His technique can be extended for set many cardinals.

# The rearrangement number

The Riemann rearrangement theorem states that a convergent real series $sum_n a_n$ is absolutely convergent if and only if the value of the sum is invariant under all rearrangements $sum_n a_{p(n)}$ by any permutation $p$ on the natural numbers; furthermore, if a series is merely conditionally convergent, then one may find rearrangements for which the new sum $sum_n a_{p(n)}$ has any desired (extended) real value or which becomes non-convergent. In recent joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson, based on an exchange in reply to a Hardy’s MathOverflow question on the topic, we investigate the minimal size of a family of permutations that can be used in this manner to test an arbitrary convergent series for absolute convergence. Specifically, we define the rearrangement number $rr$, a new cardinal characteristic of the continuum, to be the smallest cardinality of a set $P$ of permutations of the natural numbers, such that if a convergent real series $sum_n a_n$ remains convergent to the same value after any rearrangement $sum_n a_{p(n)}$ by a permutation $p$ in $P$, then it is absolutely convergent. The corresponding rearrangement number for sums, denoted rr_Sigma, is the smallest cardinality of a family $P$ of permutations, such that if a series $sum_n a_n$ is conditionally convergent, then there is some rearrangement $sum_n a_{p(n)}$, by a permutation $p$ in $P$, for which the series converges to a different value. We investigate the basic properties of these numbers, and explore their relations with other cardinal characteristics of the continuum. Our main results are that b≤ rr≤ non(M), that d≤ rr_Sigma, and that b≤ rr is relatively consistent.

# Diamond* Coding

Coding information into the structure of the universe is a forcing technique with many applications in set theory. To carry out it out, we a need a property that: i) can be easily switched on or off at (e.g.) each regular cardinal in turn, and ii) is robust with regards both to small and to highly-closed forcing. GCH coding, controlling the success or failure of the GCH at each cardinal in turn, is the most widely used, and for good reason: there are simple forcings that turn it on and off, and it is easily seen to be unaffected by small or highly-closed forcing. However, it does have limitations – most obviously, GCH coding is of necessity incompatible with the GCH itself. In this talk I will present an alternative coding using the property Diamond*, a variant of the classic Diamond. I will discuss Diamond* and demonstrate that it satisfies the requirements for coding while preserving the GCH.

Although the basic techniques for controlling Diamond* have been known for some time, to my knowledge the first use of Diamond* as a coding axiom was by Andrew Brooke-Taylor in his work on definable well-orders of the universe. I will follow the excellent exposition presented in his dissertation.

# 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.

# Force to change large cardinal strength

Suppose $kappain V$ is a cardinal with large cardinal property $A$. In this talk, I will present several theorems which exhibit a notion of forcing $mathbb P$ such that if $Gsubseteq mathbb P$ is $V$-generic, then the cardinal $kappa$ no longer has property $A$ in the forcing extension $V[G]$, but has as many large cardinal properties below $A$ as possible. I will also introduce new large cardinal notions and degrees for large cardinal properties.

This talk is the speaker’s dissertation defense.

# News on the Solid Core

Set-theoretic geology, a line of research jointly created by Hamkins, Reitz and myself, introduced some inner models which result from inverting forcing in some sense. For example, the mantle of a model of set theory V is the intersection of all inner models of which V is an extension by set-forcing. It was an initial, naive hope that one might arrive at a model that is in some sense canonical, but one of the main results on set-theoretic geology is that this is not so: every model of set theory V has a class forcing extension V[G] so that the mantle, as computed in V[G], is V. So quite literally, the mantle of a model of set theory can be anything.

In an attempt to arrive at a concept that fits in with the general spirit of set-theoretic geology, but that stands a chance of being canonical, I defined a set to be solid if it cannot be added to an inner model by set-forcing, and I termed the union of all solid sets the “solid core”.

I will present some results on the solid core which were obtained in recent joint work with Ralf Schindler, and which show that the solid core indeed is a canonical inner model, assuming large cardinals (more precisely, if there is an inner model with a Woodin cardinal), but that it is not as canonical as one might have hoped without that assumption.

# Additional remarks on remarkable cardinals

This is a continuation of the earlier Introduction to remarkable cardinals lecture. The speaker will continue to discuss the various equivalent characterizations of remarkable cardinals and their relationship to other large cardinal notions.

# Namba-like Forcings at Successors of Singular Cardinals

Following up on Peter Koepke’s Logic Workshop lecture of March 22, 2013, I will discuss Namba-like forcings which either exist or can be forced to exist at successors of singular cardinals.

# Structural Connections Between a Forcing Class and its Modal Logic

This talk is on recent work with Joel Hamkins and Benedikt Loewe on ways in which finite-frame properties of specific modal logics can be combined with assertions in ZFC to show that these modal logics are related to those which arise from interpreting Gamma-forcing extensions of a model of ZFC as possible worlds of a Kripke model, where Gamma can be any of several classes of notions of forcing.