We study the $\kappa $-Borel-reducibility of isomorphism relations of complete first-order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T’, if T is classifiable and T’ is unsuperstable, then the isomorphism of models of T’ is strictly above the isomorphism of models of T with respect to $\kappa $-Borel-reducibility.

Kolmogorov conditionalization is a strategy for updating credences based on propositions that have initial probability 0. I explore the connection between Kolmogorov conditionalization and Dutch books. Previous discussions of the connection rely crucially upon a factivity assumption: they assume that the agent updates credences based on true propositions. The factivity assumption discounts cases of misplaced certainty, i.e., cases where the agent invests credence 1 in a falsehood. Yet misplaced certainty arises routinely in scientific and philosophical applications of Bayesian decision theory. I prove a non-factive Dutch book theorem and converse Dutch book theorem for Kolmogorov conditionalization. The theorems do not rely upon the factivity assumption, so they establish that Kolmogorov conditionalization has unique pragmatic virtues that persist even in cases of misplaced certainty.

In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $\nu $ such that the singular cardinal hypothesis fails at $\nu $ and every collection of fewer than $\operatorname {\mathrm {cf}}(\nu )$ stationary subsets of $\nu ^{+}$ reflects simultaneously. For $\operatorname {\mathrm {cf}}(\nu )> \omega $, this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\operatorname {\mathrm {cf}}(\nu ) = \omega $ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$-c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $-c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$.

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3].

We consider of constructing normal ultrafilters in extensions are here Easton support iterations of Prikry-type forcing notions. New ways presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

Katok’s special representation theorem states that any free ergodic measure- preserving $\mathbb {R}^{d}$-flow can be realized as a special flow over a $\mathbb {Z}^{d}$-action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel $\mathbb {R}^{d}$-flows emerge from $\mathbb {Z}^{d}$-actions through the special flow construction using bi-Lipschitz cocycles.

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

The smooth development of large parts of mathematics hinges on the idea that some sets are ‘small’ or ‘negligible’ and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely ‘measure zero’, originated with Lebesgue, while a second smallness notion, namely ‘meagre’ or ‘first category’, originated with Baire around the same time. The associated Baire category theorem is a central result governing the properties of meagre (and related) sets, while the same holds for Tao’s pigeonhole principle for measure spaces and measure zero sets. In this paper, we study these theorems in Kohlenbach’s higher-order Reverse Mathematics, identifying a considerable number of equivalent and robust theorems. The latter involve most basic properties of semi-continuous and pointwise discontinuous functions, Blumberg’s theorem, Riemann integration, and Volterra’s early work circa 1881. All the aforementioned theorems fall (far) outside of the Big Five of Reverse Mathematics, and we investigate natural restrictions like Baire 1 and quasi-continuity that make these theorems provable again in the Big Five (or similar). Finally, despite the fundamental differences between measure and category, the proofs of our equivalences turn out to be similar.

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, Chapter 3].

We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of the ground model). One direction is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed. This work is a part of my upcoming thesis [7].

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

This paper extends the investigations into logical properties of the quantified argument calculus (Quarc) by suggesting a series of proper subsystems which, although retaining the entire vocabulary of Quarc, restrict quantification in such a way as to make the result decidable. The proof of decidability is via a procedure that prunes the infinite branches of a derivation tree in what is a syntactic counterpart of semantic filtration. We demonstrate an application of one of these systems by showing that Aristotle’s assertoric syllogistic is embeddable within, thus also providing another method of showing its decidability.

This paper puts forth a class of algebraic structures, relativized Boolean algebras (RBAs), that provide semantics for propositional logic in which truth/validity is only defined relative to a local domain. In particular, the join of an event and its complement need not be the top element. Nonetheless, behavior is locally governed by the laws of propositional logic. By further endowing these structures with operators—akin to the theory of modal Algebras—RBAs serve as models of modal logics in which truth is relative. In particular, modal RBAs provide semantics for various well-known awareness logics and an alternative view of possibility semantics.

The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $\mathbf {RCA}_0$, by giving a new proof of $\Sigma ^0_2$-induction.

Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems).

A propositional version of the construction shows that at least one of the following three statements is true:

1. 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm).

2. 2. $E \not \subseteq P/poly$.

3. 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${\text {NP}}$ sets.

A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $-categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$.