We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of ${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

We argue that some of Brouwer’s assumptions, rejected by Bishop, should be considered and studied as possible axioms. We show that Brouwer’s Continuity Principle enables one to prove an intuitionistic Borel Hierarchy Theorem. We also explain that Brouwer’s Fan Theorem is useful for a development of the theory of measure and integral different from the one worked out by Bishop. We show that Brouwer’s bar theorem not only proves the Fan Theorem but also a stronger statement that we call the Almost-fan Theorem. The Almost-fan Theorem implies intuitionistic versions of Ramsey’s Theorem and the Bolzano-Weierstrass Theorem.

We present a streamlined and simplified exponential lower bound on the length of proofs in intuitionistic implicational logic, adapted to Gordeev and Haeusler’s dag-like natural deduction.

Heyting’s intuitionistic predicate logic describes very general regularities observed in constructive mathematics. The intended meaning of the logical constants is clarified through Heyting’s proof interpretation. A re-evaluation of proof interpretation and predicate logic leads to the new constructive Basic logic properly contained in intuitionistic logic. We develop logic and interpretation simultaneously by an axiomatic approach. Basic logic appears to be complete. A brief historical overview shows that our insights are not all new.

Many-valued logics, in general, and real-valued logics, in particular, usually focus on a notion of consequence based on preservation of full truth, typically represented by the value $1$ in the semantics given in the real unit interval $[0,1]$. In a recent paper [Foundations of Reasoning with Uncertainty via Real-valued Logics, Proceedings of the National Academy of Sciences 121(21): e2309905121, 2024], Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on a rich class of sentences, multi-dimensional sentences, that talk about combinations of any possible truth values of real-valued formulas. They have proved a strong completeness result that allows one to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. In this paper, we extend that work to the first-order (as well as modal) logic of multi-dimensional sentences. We give a parameterized axiomatic system that covers any reasonable logic and prove a corresponding completeness theorem, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a zero-one law for finitely-valued versions of these logics. Since several first-order real-valued logics are known not to have recursive axiomatizations but only infinitary ones, our system is by force akin to infinitary systems.

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory can be given an analytically true reading in the logical framework.

In this thesis, we study the complexity of theorems that may be considered partially impredicative from the point of view of reverse mathematics and Weihrauch degrees.

From the perspective of reverse mathematics and ordinal analysis, the axiomatic system $\mathsf {ATR}_0$ is known as the limit of predicativity, and $\Pi ^1_{1}\text {-}\mathsf {CA}_0$ is known as an impredicative system. In this thesis, we study the complexity of some theorems that are stronger than $\mathsf {ATR}_0$ and weaker than $\Pi ^1_{1}\text {-}\mathsf {CA}_0$ from the point of view of reverse mathematics and Weihrauch degrees.

In Chapter 3, we study some problems related to Knaster–Tarski’s theorem. Knaster–Tarski’s theorem states that any monotone operator on $2^{\omega }$ has a least fixed point. Avigad introduced a weaker variant, $\mathsf {FP}$, which asserts the existence of a fixed point instead of the least fixed point, and proved that $\mathsf {FP}$ for arithmetical operators is equivalent to $\mathsf {ATR}_0$ over $\mathsf {RCA}_0$. In this thesis, we show that $\mathsf {FP}$ for $\Sigma ^0_2$-operators is strictly stronger than $\mathsf {ATR}_2$, a Weihrauch degree corresponding to $\mathsf {ATR}_0$, in terms of Weihrauch reduction. In addition, we study the bottom-up proof of Knaster–Tarski’s theorem. It is known that the least fixed point of a monotone operator is given by the $\omega _1$-times iteration of the operator at the empty set. This implies that any monotone operator involves a hierarchy formed by the iterative applications of the operator, starting with the empty set and reaching the least fixed point. We prove that although the existence of a hierarchy is equivalent to $\mathsf {ATR}_0$ over $\mathsf {ACA}_0$, it is stronger than $\mathsf {C}_{\omega ^{\omega }}$ in the terms of Weirhauch reduction.

In Chapter 5, we study the relative leftmost path principle in Weihrauch degrees. This principle was introduced by Towsner to study partial impredicativity in reverse mathematics. He gave a hierarchy between $\mathsf {ATR}_0$ and $\Pi ^1_1\text {-}\mathsf {CA}_0$ by this principle. We show that this principle also makes a hierarchy between $\mathsf {ATR}_2$ and $\mathsf {C}_{\omega ^{\omega }}$ in Weihrauch degrees. We also show that the relative leftmost path principle is, not the same as, but very close to a variant of $\beta $-model reflection.

In Chapter 6, we introduce a hierarchy dividing $\{\sigma \in \Pi ^1_2 : \Pi ^1_1\text {-}\mathsf {CA}_0 \vdash \sigma \}$. Then, we give some characterizations of this hierarchy using some principles equivalent to $\Pi ^1_1\text {-}\mathsf {CA}_0$: leftmost path principle, Ramsey’s theorem for $\Sigma ^0_n$ classes of $[\mathbb {N}]^{\mathbb {N}}$ and the determinacy of Gale–Stewart game for $(\Sigma ^0_1)_n$ classes. As an application, our hierarchy explicitly shows that the number of applications of the hyperjump operator needed to prove $\Sigma ^0_n$ Ramsey’s theorem or $(\Sigma ^0_1)_n$ determinacy increases when the subscript n increases.

Abstract prepared by Yudai Suzuki.

E-mail: yudai.suzuki.research@gmail.com.

Arithmetic and logic seem to enjoy an especially close relationship. Frege once wrote that arithmetic is reason’s nearest kin. To deny any of the basic laws of arithmetic seems tantamount to denying a basic law of logic. My dissertation is concerned with two great attempts to make something more of this informal idea. In one direction, Frege tried to reduce arithmetic to nothing but quantificational logic and definitions. Neologicists continue to follow in Frege’s footsteps, pursuing a version of this program today. In the other direction, Gödel tried to reduce certain applications of quantificational logic to nothing but arithmetic and definitions, by means of his Dialectica translation.

In the first half of my dissertation, I prove new theorems (with Jeremy Avigad) that shed a surprising light on the prospects for neologicism. An important objection against neologicism is that it makes use of allegedly stipulative definitions that are not conservative over pure logic, i.e., definitions that yield new consequences expressible in old vocabulary. This violates a basic requirement on stipulative definitions. I argue that by passing to a richer logical and definitional framework, it is possible to overcome the conservativeness objection. However, there is a subtlety: the strategy succeeds only if conservativeness is understood semantically rather than deductively. This suggests that the viability of neologicism is highly sensitive to the way in which epistemic commitments are represented in formal theories.

In the second half of my dissertation, I argue that Gödel’s Dialectica translation succeeds in assigning a constructive meaning to quantificational theories of arithmetic. Virtually all commentators have objected that Gödel’s translation makes use of definitions which presuppose the very quantificational logic that Gödel was trying to eliminate. This would render the translation philosophically circular. Gödel was adamant that there was no circularity here. He attempted to explain the matter in a page-long footnote, which, however, no one has been able to understand. I vindicate Gödel, showing that there is no circularity and answering a longstanding exegetical question in Gödel scholarship.

Abstract prepared by Stephen Mackereth

E-mail: sgmackereth@gmail.com

URL: http://d-scholarship.pitt.edu/id/eprint/46578