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We prove several consistency results concerning the notion of
$\omega $
-strongly measurable cardinal in
$\operatorname {\mathrm {HOD}}$
. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than
$o(\kappa ) = \kappa $
, that every successor of a regular cardinal is
$\omega $
-strongly measurable in
$\operatorname {\mathrm {HOD}}$
.
Generalizing the notion of a tight almost disjoint family, we introduce the notions of a tight eventually different family of functions in Baire space and a tight eventually different set of permutations of
$\omega $
. Such sets strengthen maximality, exist under
$\mathsf {MA} (\sigma \mathrm {-centered})$
and come with a properness preservation theorem. The notion of tightness also generalizes earlier work on the forcing indestructibility of maximality of families of functions. As a result we compute the cardinals
$\mathfrak {a}_e$
and
$\mathfrak {a}_p$
in many known models by giving explicit witnesses and therefore obtain the consistency of several constellations of cardinal characteristics of the continuum including
$\mathfrak {a}_e = \mathfrak {a}_p = \mathfrak {d} < \mathfrak {a}_T$
,
$\mathfrak {a}_e = \mathfrak {a}_p < \mathfrak {d} = \mathfrak {a}_T$
,
$\mathfrak {a}_e = \mathfrak {a}_p =\mathfrak {i} < \mathfrak {u}$
, and
$\mathfrak {a}_e=\mathfrak {a}_p = \mathfrak {a} < non(\mathcal N) = cof(\mathcal N)$
. We also show that there are
$\Pi ^1_1$
tight eventually different families and tight eventually different sets of permutations in L thus obtaining the above inequalities alongside
$\Pi ^1_1$
witnesses for
$\mathfrak {a}_e = \mathfrak {a}_p = \aleph _1$
.
Moreover, we prove that tight eventually different families are Cohen indestructible and are never analytic.
There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou’s probabilistic bisimilarity, whereas the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator
$\mathcal {O}$
, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the Zhou ordinal of
$\mathbb {S}$
,
$\mathfrak {Z}(\mathbb {S})$
. When
$\mathfrak {Z}(\mathbb {S})=0$
,
$\mathbb {S}$
satisfies the Hennessy–Milner property. The second author proved the existence of an LMP
$\mathbb {S}$
with
$\mathfrak {Z}(\mathbb {S}) \geq 1$
and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs
$\mathbb {S}$
over separable metrizable spaces having arbitrary large countable
$\mathfrak {Z}(\mathbb {S})$
and that it is consistent with the axioms of
$\mathit {ZFC}$
that there is such a process with an uncountable Zhou ordinal.
Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic
$\mathbf {GL}$
is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.
Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in
$\mathsf {ZF}$
, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.
After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter.
In this paper, we characterize the possible cofinalities of the least
$\lambda $
-strongly compact cardinal. We show that, on the one hand, for any regular cardinal,
$\delta $
, that carries a
$\lambda $
-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above
$\delta $
, that the least
$\lambda $
-strongly compact cardinal has cofinality
$\delta $
. On the other hand, provably the cofinality of the least
$\lambda $
-strongly compact cardinal always carries a
$\lambda $
-complete uniform ultrafilter.
A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.
We investigate iterating the construction of
$C^{*}$
, the L-like inner model constructed using first order logic augmented with the “cofinality
$\omega $
” quantifier. We first show that
$\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$
is equiconsistent with
$\mathrm {ZFC}$
, as well as having finite strictly decreasing sequences of iterated
$C^{*}$
s. We then show that in models of the form
$L[U]$
we get infinite decreasing sequences of length
$\omega $
, and that an inner model with a measurable cardinal is required for that.
In the Zermelo–Fraenkel set theory with the Axiom of Choice, a forcing notion is “
$\kappa $
-distributive” if and only if it is “
$\kappa $
-sequential.” We show that without the Axiom of Choice, this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for
$\kappa $
. Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that although a
$\kappa $
-distributive forcing notion may violate Dependent Choice, it must preserve the Axiom of Choice for families of size
$\kappa $
. On the other hand, a
$\kappa $
-sequential can violate the Axiom of Choice for countable families. We also provide a condition of “quasiproperness” which is sufficient for the preservation of Dependent Choice, and is also necessary if the forcing notion is sequential.
We construct a nonseparable Banach space
$\mathcal {X}$
(actually, of density continuum) such that any uncountable subset
$\mathcal {Y}$
of the unit sphere of
$\mathcal {X}$
contains uncountably many points distant by less than
$1$
(in fact, by less then
$1-\varepsilon $
for some
$\varepsilon>0$
). This solves in the negative the central problem of the search for a nonseparable version of Kottman’s theorem which so far has produced many deep positive results for special classes of Banach spaces and has related the global properties of the spaces to the distances between points of uncountable subsets of the unit sphere. The property of our space is strong enough to imply that it contains neither an uncountable Auerbach system nor an uncountable equilateral set. The space is a strictly convex renorming of the Johnson–Lindenstrauss space induced by an
$\mathbb {R}$
-embeddable almost disjoint family of subsets of
$\mathbb {N}$
. We also show that this special feature of the almost disjoint family is essential to obtain the above properties.
After reviewing various natural bi-interpretations in urelement set theory, including second-order set theories with urelements, we explore the strength of second-order reflection in these contexts. Ultimately, we prove that second-order reflection with the abundant atom axiom is bi-interpretable and hence also equiconsistent with the existence of a supercompact cardinal. The proof relies on a reflection characterization of supercompactness, namely, a cardinal
$\kappa $
is supercompact if and only if every
$\Pi ^1_1$
sentence true in a structure M (of any size) containing
$\kappa $
in a language of size less than
$\kappa $
is also true in a substructure
$m\prec M$
of size less than
$\kappa $
with
$m\cap \kappa \in \kappa $
.
We construct a model of set theory in which there exists a Suslin tree and satisfies that any two normal Aronszajn trees, neither of which contains a Suslin subtree, are club isomorphic. We also show that if S is a free normal Suslin tree, then for any positive integer n there is a c.c.c. forcing extension in which S is n-free but all of its derived trees of dimension greater than n are special.
We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially
$\boldsymbol {\Pi }^{0}_{\alpha + 1}$
isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank
$\alpha $
,
$\alpha \geq 2$
. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures.
We prove that the category
$\mathsf {SBor}$
of standard Borel spaces is the (bi-)initial object in the 2-category of countably complete Boolean (countably) extensive categories. This means that
$\mathsf {SBor}$
is the universal category admitting some familiar algebraic operations of countable arity (e.g., countable products and unions) obeying some simple compatibility conditions (e.g., products distribute over disjoint unions). More generally, for any infinite regular cardinal
$\kappa $
, the dual of the category
$\kappa \mathsf {Bool}_{\kappa }$
of
$\kappa $
-presented
$\kappa $
-complete Boolean algebras is (bi-)initial in the 2-category of
$\kappa $
-complete Boolean (
$\kappa $
-)extensive categories.
We give a new characterization of the cardinal invariant
$\mathfrak {d}$
as the minimal cardinality of a family
$\mathcal {D}$
of tall summable ideals such that an ultrafilter is rapid if and only if it has non-empty intersection with all the ideals in the family
$\mathcal {D}$
. On the other hand, we prove that in the Miller model, given any family
$\mathcal {D}$
of analytic tall p-ideals such that
$\vert \mathcal {D}\vert <\mathfrak {d}$
, there is an ultrafilter
$\mathcal {U}$
which is an
$\mathscr {I}$
-ultrafilter for all ideals
$\mathscr {I}\in \mathcal {D}$
at the same time, yet
$\mathcal {U}$
is not a rapid ultrafilter. As a corollary, we obtain that in the Miller model, given any analytic tall p-ideal
$\mathscr {I}$
,
$\mathscr {I}$
-ultrafilters are dense in the Rudin–Blass ordering, generalizing a theorem of Bartoszyński and S. Shelah, who proved that in such model, Hausdorff ultrafilters are dense in the Rudin–Blass ordering. This theorem also shows some limitations about possible generalizations of a theorem of C. Laflamme and J. Zhu.
Boolean-valued models generalize classical two-valued models by allowing arbitrary complete Boolean algebras as value ranges. The goal of my dissertation is to study Boolean-valued models and explore their philosophical and mathematical applications.
In Chapter 1, I build a robust theory of first-order Boolean-valued models that parallels the existing theory of two-valued models. I develop essential model-theoretic notions like “Boolean-valuation,” “diagram,” and “elementary diagram,” and prove a series of theorems on Boolean-valued models, including the (strengthened) Soundness and Completeness Theorem, the Löwenheim–Skolem Theorems, the Elementary Chain Theorem, and many more.
Chapter 2 gives an example of a philosophical application of Boolean-valued models. I apply Boolean-valued models to the language of mereology to model indeterminacy in the parthood relation. I argue that Boolean-valued semantics is the best degree-theoretic semantics for the language of mereology. In particular, it trumps the well-known alternative—fuzzy-valued semantics. I also show that, contrary to what many have argued, indeterminacy in parthood entails neither indeterminacy in existence nor indeterminacy in identity, though being compatible with both.
Chapter 3 (joint work with Bokai Yao) gives an example of a mathematical application of Boolean-valued models. Scott and Solovay famously used Boolean-valued models on set theory to obtain relative consistency results. In Chapter 3, I investigate two ways of extending the Scott–Solovay construction to set theory with urelements. I argue that the standard way of extending the construction faces a serious problem, and offer a new way that is free from the problem.
Assume
$\mathsf {ZF} + \mathsf {AD}$
and all sets of reals are Suslin. Let
$\Gamma $
be a pointclass closed under
$\wedge $
,
$\vee $
,
$\forall ^{\mathbb {R}}$
, continuous substitution, and has the scale property. Let
$\kappa = \delta (\Gamma )$
be the supremum of the length of prewellorderings on
$\mathbb {R}$
which belong to
$\Delta = \Gamma \cap \check \Gamma $
. Let
$\mathsf {club}$
denote the collection of club subsets of
$\kappa $
. Then the countable length everywhere club uniformization holds for
$\kappa $
: For every relation
$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$
with the property that for all
$\ell \in {}^{<{\omega _1}}\kappa $
and clubs
$C \subseteq D \subseteq \kappa $
,
$R(\ell ,D)$
implies
$R(\ell ,C)$
, there is a uniformization function
$\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$
with the property that for all
$\ell \in \mathrm {dom}(R)$
,
$R(\ell ,\Lambda (\ell ))$
. In particular, under these assumptions, for all
$n \in \omega $
,
$\boldsymbol {\delta }^1_{2n + 1}$
satisfies the countable length everywhere club uniformization.
We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership
$x\in y\in x$
. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.