Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $, we do not in general know how to characterize the degrees $\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$ below which L can be bounded. The important characterizations known are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf {d}$ if and only if $\mathbf {d}$ contains sets of “fickleness” $>\omega $ and $\geq \omega ^\omega $ respectively. We work towards finding a lattice that characterizes the levels above $\omega ^2$, the first non-trivial level after $\omega $. We introduced a lattice-theoretic property called “$3$-directness” to describe lattices that are no “wider” or “taller” than $L_7$ and $M_3$. We exhaust the 3-direct lattices L, but they turn out to also characterize the $>\omega $ or $\geq \omega ^\omega $ levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides $M_3$ that also characterize the $\geq \omega ^\omega $-levels. Our search for a $>\omega ^2$-candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq \omega ^\omega $-lattices as sublattices.

Abstract prepared by Liling Ko.

E-mail: ko.390@osu.edu

URL: http://sites.nd.edu/liling-ko/

We devise imperative programming languages for verified real number computation where real numbers are provided as abstract data types such that the users of the languages can express real number computation by considering real numbers as abstract mathematical entities. Unlike other common approaches toward real number computation, based on an algebraic model that lacks implementability or transcendental computation, or finite-precision approximation such as using double precision computation that lacks a formal foundation, our languages are devised based on computable analysis, a foundation of rigorous computation over continuous data. Consequently, the users of the language can easily program real number computation and reason about the behaviours of their programs, relying on their mathematical knowledge of real numbers without worrying about artificial roundoff errors. As the languages are imperative, we adopt precondition–postcondition-style program specification and Hoare-style program verification methodologies. Consequently, the users of the language can easily program a computation over real numbers, specify the expected behaviour of the program, including termination, and prove the correctness of the specification. Furthermore, we suggest extending the languages with other interesting continuous data, such as matrices, continuous real functions, et cetera.

Abstract taken directly from the thesis.

E-mail: sewonpark17@gmail.com

URL: https://sewonpark.com/thesis

The dissertation introduces new sequent-calculi for free first- and second-order logic, and a hyper-sequent calculus for modal logics K, D, T, B, S4, and S5; to attain the calculi for the stronger modal logics, only external structural rules need to be added to the calculus for K, while operational and internal structural rules remain the same. Completeness and cut-elimination are proved for all calculi presented.

Philosophically, the dissertation develops an inferentialist, or proof-theoretic, theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. The dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators.

Abstract prepared by Andrew Parisi

E-mail: andrew.p.parisi@gmail.com

URL: https://opencommons.uconn.edu/dissertations/1480/

The thin set theorem for n-tuples and k colors ($\operatorname {\mathrm {\sf {TS}}}^n_k$) states that every k-coloring of $[\mathbb {N}]^n$ admits an infinite set of integers H such that $[H]^n$ avoids at least one color. In this paper, we study the combinatorial weakness of the thin set theorem in reverse mathematics by proving neither $\operatorname {\mathrm {\sf {TS}}}^n_k$, nor the free set theorem ($\operatorname {\mathrm {\sf {FS}}}^n$) imply the Erdős–Moser theorem ($\operatorname {\mathrm {\sf {EM}}}$) whenever k is sufficiently large (answering a question of Patey and giving a partial result towards a question of Cholak Giusto, Hirst and Jockusch). Given a problem $\mathsf {P}$, a computable instance of $\mathsf {P}$ is universal iff its solution computes a solution of any other computable $\mathsf {P}$-instance. It has been established that most of Ramsey-type problems do not have a universal instance, but the case of Erdős–Moser theorem remained open so far. We prove that Erdős–Moser theorem does not admit a universal instance (answering a question of Patey).

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

A celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.

We prove two theorems concerning indestructibility properties of the first two strongly compact cardinals when these cardinals are in addition the first two measurable cardinals. Starting from two supercompact cardinals $\kappa _1 < \kappa _2$, we force and construct a model in which $\kappa _1$ and $\kappa _2$ are both the first two strongly compact and first two measurable cardinals, $\kappa _1$’s strong compactness is fully indestructible (i.e., $\kappa _1$’s strong compactness is indestructible under arbitrary $\kappa _1$-directed closed forcing), and $\kappa _2$’s strong compactness is indestructible under $\mathrm {Add}(\kappa _2, \delta )$ for any ordinal $\delta $. This provides an answer to a strengthened version of a question of Sargsyan found in [17, Question 5]. We also investigate indestructibility properties that may occur when the first two strongly compact cardinals are not only the first two measurable cardinals, but also exhibit nontrivial degrees of supercompactness.

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension 1 which is not definably compact. In a future paper we will generalize this to non-abelian G.

For a ring R, Hilbert’s Tenth Problem $HTP(R)$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP(\mathbb {Z}[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$-reduce to $HTP(R_W)$. In contrast, we show that every Turing degree contains a set W for which such a $1$-reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb {Z}[W^{-1}])$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$.

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

The cardinal invariant $\mathfrak {hm}$ is defined as the minimum size of a family of $\mathsf {c}_{\mathsf {min}}$-monochromatic sets that cover $2^{\omega }$ (where $\mathsf {c}_{\mathsf {min}}( x,y) $ is the parity of the biggest initial segment both x and y have in common). We prove that $\mathfrak {hm}=\omega _{1}$ holds in Shelah’s model of $\mathfrak {i<u},$ so the inequality $\mathfrak {hm<u}$ is consistent with the axioms of $\mathsf {ZFC}$. This answers a question of Thilo Weinert. We prove that the diamond principle $\mathfrak {\Diamond }_{\mathfrak {d}}$ also holds in that model.

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

We show that it is independent whether club $\kappa $-Miller forcing preserves $\kappa ^{++}$. We show that under $\kappa ^{<\kappa }> \kappa $, club $\kappa $-Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $. Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $-Miller forcing does not have the Laver property.

Ramsey algebras are an attempt to investigate Ramsey spaces generated by algebras in a purely combinatorial fashion. Previous studies have focused on the basic properties of Ramsey algebras and a few specific examples. In this article, we study the properties of Ramsey algebras from a structural point of view. For instance, we will see that isomorphic algebras have the same Ramsey algebraic properties, but elementarily equivalent algebras need not be so, as expected. We also answer an open question about Cartesian products of Ramsey algebras.

In set theory without the Axiom of Choice ($\mathsf {AC}$), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

We ask when, for a pair of structures $\mathcal {A}_1,\mathcal {A}_2$, there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of $\mathcal {A}_1$. We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings $\mathcal {A}_1$ of type $\omega _1^{CK}$ and $\mathcal {A}_2$ of Harrison type, there should not be any such procedure, but, in fact, there is one. We construct an example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics.

We will show that almost all nonassociative relation algebras are symmetric and integral (in the sense that the fraction of both labelled and unlabelled structures that are symmetric and integral tends to $1$), and using a Fraïssé limit, we will establish that the classes of all atom structures of nonassociative relation algebras and relation algebras both have $0$–$1$ laws. As a consequence, we obtain improved asymptotic formulas for the numbers of these structures and broaden some known probabilistic results on relation algebras.