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We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals.
To exemplify: we prove that for every inaccessible cardinal
$\kappa $, if
$\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation
$\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group
$(G,+)$ of size
$\kappa $, there exists a map
$f:G\rightarrow G$ such that for every
$X\subseteq G$ of size
$\kappa $ and every
$g\in G$, there exist
$x\neq y$ in X such that
$f(x+y)=g$.
We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.
This article deals with the problem of when, given a collection
$\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in
$\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to
$L_1[0,1]$ factors through Z.
We also prove the following descriptive set theoretical result: Let
$\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if
$\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for
$A \in \mathcal {B}$, the assignment
$A \to A^*$ can be realised by a Borel map
$\mathcal {B}\to \mathcal {L}$.
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.
We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation
$(X, E)$
may be realized as the topological ergodic decomposition of a continuous action of a countable group
$\Gamma \curvearrowright X$
generating E. We then apply this to the study of the cardinal algebra
$\mathcal {K}(E)$
of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation
$(X, E)$
. We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.
This paper critically examines two arguments against the generic multiverse, both of which are due to W. Hugh Woodin. Versions of the first argument have appeared a number of times in print, while the second argument is relatively novel. We shall investigate these arguments through the lens of two different attitudes one may take toward the methodology and metaphysics of set theory; and we shall observe that the impact of these arguments depends significantly on which of these attitudes is upheld. Our examination of the second argument involves the development of a new (inner) model for Steel’s multiverse theory, which is delivered in the Appendix.
A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.
We isolate two abstract determinacy theorems for games of length
$\omega_1$
from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals that
(1) if the Continuum Hypothesis holds, then all games of length
$\omega_1$
which are provably
$\Delta_1$
-definable from a universally Baire parameter (in first-order or
$\Omega $
-logic) are determined;
(2) all games of length
$\omega_1$
with payoff constructible relative to the play are determined; and
(3) if the Continuum Hypothesis holds, then there is a model of
${\mathsf{ZFC}}$
containing all reals in which all games of length
$\omega_1$
definable from real and ordinal parameters are determined.
We prove the Decomposability Conjecture for functions of Baire class
$2$
from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.
We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not
$\sigma $
-continuous with closed witnesses.
A set
$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$
is universal for countable subsets of
${\mathbb {R}}$
if and only if for all
$x \in {\mathbb {R}}$
, the section
$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$
is countable and for all countable sets
$A \subseteq {\mathbb {R}}$
, there is an
$x \in {\mathbb {R}}$
so that
$U_x = A$
. Define the equivalence relation
$E_U$
on
${\mathbb {R}}$
by
$x_0 \ E_U \ x_1$
if and only if
$U_{x_0} = U_{x_1}$
, which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump,
$=^+$
, of the equality relation takes the form
$E_{U^*}$
where
$U^*$
is the most natural Borel set that is universal for countable sets. The main result is that
$=^+$
and
$E_U$
for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets,
$E_U$
is Borel bireducible to
$=^+$
. If one assumes a particular instance of
$\mathbf {\Sigma }_3^1$
-generic absoluteness, then for all
$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$
that are
$\mathbf {\Sigma }_1^1$
(continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of
$=^+$
into
$E_U$
.
The class forcing theorem, which asserts that every class forcing notion
${\mathbb {P}}$
admits a forcing relation
$\Vdash _{\mathbb {P}}$
, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory
$\text {GBC}$
to the principle of elementary transfinite recursion
$\text {ETR}_{\text {Ord}}$
for class recursions of length
$\text {Ord}$
. It is also equivalent to the existence of truth predicates for the infinitary languages
$\mathcal {L}_{\text {Ord},\omega }(\in ,A)$
, allowing any class parameter A; to the existence of truth predicates for the language
$\mathcal {L}_{\text {Ord},\text {Ord}}(\in ,A)$
; to the existence of
$\text {Ord}$
-iterated truth predicates for first-order set theory
$\mathcal {L}_{\omega ,\omega }(\in ,A)$
; to the assertion that every separative class partial order
${\mathbb {P}}$
has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most
$\text {Ord}+1$
. Unlike set forcing, if every class forcing notion
${\mathbb {P}}$
has a forcing relation merely for atomic formulas, then every such
${\mathbb {P}}$
has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between
$\text {GBC}$
and Kelley–Morse set theory
$\text {KM}$
.
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.
We characterize the determinacy of
$F_\sigma $
games of length
$\omega ^2$
in terms of determinacy assertions for short games. Specifically, we show that
$F_\sigma $
games of length
$\omega ^2$
are determined if, and only if, there is a transitive model of
${\mathsf {KP}}+{\mathsf {AD}}$
containing
$\mathbb {R}$
and reflecting
$\Pi _1$
facts about the next admissible set.
As a consequence, one obtains that, over the base theory
${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$
exists,” determinacy for
$F_\sigma $
games of length
$\omega ^2$
is stronger than
${\mathsf {AD}}$
, but weaker than
${\mathsf {AD}} + \Sigma _1$
-separation.
Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to
$\mathsf {DC}$
-preserving symmetric submodels of forcing extensions. Hence,
$\mathsf {ZF}+\mathsf {DC}$
not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in
$\mathsf {ZF}$
, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in
$\mathsf {ZF}+\mathsf {DC}$
and
$\mathsf {ZFC}$
. Our results confirm
$\mathsf {ZF} + \mathsf {DC}$
as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.
This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.
For a given inner model N of ZFC, one can consider the relativized version of Berkeley cardinals in the context of ZFC, and ask if there can exist an “N-Berkeley cardinal.” In this article we provide a positive answer to this question. Indeed, under the assumption of a supercompact cardinal
$\delta $
, we show that there exists a ZFC inner model N such that there is a cardinal which is N-Berkeley, even in a strong sense. Further, the involved model N is a weak extender model of
$\delta $
is supercompact. Finally, we prove that the strong version of N-Berkeley cardinals turns out to be inconsistent whenever N satisfies closure under
$\omega $
-sequences.
We study a partial order on countably complete ultrafilters introduced by Ketonen [2] as a generalization of the Mitchell order. The following are our main results: the order is wellfounded; its linearity is equivalent to the Ultrapower Axiom, a principle introduced in the author’s dissertation [1]; finally, assuming the Ultrapower Axiom, the Ketonen order coincides with Lipschitz reducibility in the sense of generalized descriptive set theory.