The distinction of the semantic spaces of elements and types is common practice in practically all type systems. A few type systems, including some early ones, have been proposed whose semantic space has functions only, i.e., depending on the context functions may play element roles as well as type roles. All of these systems are either lacking expressive power, in particular, polymorphism, or they violate uniqueness of types. This work presents for the first time a function-based type system in which typing is a relation between functions and which is using an ordering of functions to introduce bounded polymorphism. The ordering is based on an infinite set of top objects, itself strictly linearly ordered, each of which characterizes a certain function space. These top objects are predicative in the sense that a function using some top object cannot be smaller than this object. The interpretation of proposition as types and elements as proofs remains valid and is extended by viewing the ordering between types as logical implication. The proposed system can be shown to satisfy confluence and subject reduction. Furthermore one can show that the ordering is a partial order, every set of expressions has a maximal element, and there is a (unique) minimal, logically strongest, type among all types of an element. The latter result implies an alternative notion of uniqueness of types. Strong normalisation is the deepest property and its proof is based on a well-founded relation defined over a subsystem of expressions without eliminators. Semantic abstraction of the objects involved in typing, i.e., to use functions in element as well as type roles in a relational setting, is the major contribution of function-based type systems. This work shows that dependent products are not necessary for defining type systems with bounded polymorphism, rather it presents a consistent system with bounded polymorphism and minimal types where typing is a relation between partially ordered functions.

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set of objects whatsoever. In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation.

A left-variable word over an alphabet A is a word over $A \cup \{\star \}$ whose first letter is the distinguished symbol $\star $ standing for a placeholder. The ordered variable word theorem ($\mathsf {OVW}$), also known as Carlson–Simpson’s theorem, is a tree partition theorem, stating that for every finite alphabet A and every finite coloring of the words over A, there exists a word $c_0$ and an infinite sequence of left-variable words $w_1, w_2, \dots $ such that $\{ c_0 \cdot w_1[a_1] \cdot \dots \cdot w_k[a_k] : k \in \mathbb {N}, a_1, \dots , a_k \in A \}$ is monochromatic.

In this article, we prove that $\mathsf {OVW}$ is $\Pi ^0_4$-conservative over $\mathsf {RCA}_0 + \mathsf {B}\Sigma ^0_2$. This implies in particular that $\mathsf {OVW}$ does not imply $\mathsf {ACA}_0$ over $\mathsf {RCA}_0$. This is the first principle for which the only known separation from $\mathsf {ACA}_0$ involves non-standard models.

We consider equational theories based on axioms for recursively defining functions, with rules for equality and substitution, but no form of induction—we denote such equational theories as PETS for pure equational theories with substitution. An example is Cook’s system PV without its rule for induction. We show that the Bounded Arithmetic theory $\mathrm {S}^{1}_2$ proves the consistency of PETS. Our approach employs model-theoretic constructions for PETS based on approximate values resembling notions from domain theory in Bounded Arithmetic, which may be of independent interest.

The distinction between the proofs that only certify the truth of their conclusion and those that also display the reasons why their conclusion holds has a long philosophical history. In the contemporary literature, the grounding relation—an objective, explanatory relation which is tightly connected with the notion of reason—is receiving considerable attention in several fields of philosophy. While much work is being devoted to characterising logical grounding in terms of deduction rules, no in-depth study focusing on the difference between grounding rules and logical rules exists. In this work, we analyse the relation between logical grounding and classical logic by focusing on the technical and conceptual differences that distinguish grounding rules and logical rules. The calculus employed to conduct the analysis provides moreover a strong confirmation of the fact that grounding derivations are logical derivations of a certain kind, without trivialising the distinction between grounding and logical rules, explanatory and non-explanatory parts of a derivation. By a further formal analysis, we negatively answer the question concerning the possible correspondence between grounding rules and intuitionistic logical rules.

The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every countable recursively saturated model of $\mathsf {PA}$ (Peano arithmetic) carries a full satisfaction class. This result implies that the compositional theory of truth over $\mathsf {PA}$ commonly known as $\mathsf {CT}^{-}[\mathsf {PA}]$ is conservative over $\mathsf {PA}$. In contrast, Pakhomov and Enayat (2019) showed that the addition of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true iff one of its disjuncts is true) to $\mathsf {CT}^{-}[\mathsf {PA}]$ axiomatizes the theory of truth $\mathsf {CT}_{0}[\mathsf {PA}]$ that was shown by Wcisło and Łełyk (2017) to be nonconservative over $\mathsf {PA}$. The main result of this paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full satisfaction classes over arbitrary countable recursively saturated models of $\mathsf {PA}$ that satisfy arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its approximations.

In this paper, we study the employment of $\Sigma _1$-sentences with certificates, i.e., $\Sigma _1$-sentences where a number of principles is added to ensure that the witness is sufficiently number-like. We develop certificates in some detail and illustrate their use by reproving some classical results and proving some new ones. An example of such a classical result is Vaught’s theorem of the strong effective inseparability of $\mathsf {R}_0$.

We also develop the new idea of a theory being $\mathsf {R}_{0\mathsf {p}}$-sourced. Using this notion, we can transfer a number of salient results from $\mathsf {R}_0$ to a variety of other theories.

Reverse Mathematics (RM) is a program in the foundations of mathematics where the aim is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. Generally, the minimal axioms are equivalent to the theorem at hand, assuming a weak logical system called the base theory. Moreover, many theorems are either provable in the base theory or equivalent to one of four logical systems, together called the Big Five. For instance, the Weierstrass approximation theorem, i.e., that a continuous function can be approximated uniformly by a sequence of polynomials, has been classified in RM as being equivalent to weak König’s lemma, the second Big Five system. In this paper, we study approximation theorems for discontinuous functions via Bernstein polynomials from the literature. We obtain many equivalences between the latter and weak König’s lemma. We also show that slight variations of these approximation theorems fall far outside of the Big Five but fit in the recently developed RM of new ‘big’ systems, namely the uncountability of ${\mathbb R}$, the enumeration principle for countable sets, the pigeon-hole principle for measure, and the Baire category theorem.

We show that there exists a recursive coloring function c such that any $\Pi ^0_3$ set is not a solution to c for Hindman’s theorem. We also show that there exists a recursive coloring function c such that any $\Delta ^0_3$ set is not a solution to c for Hindman’s theorem restricted to sums of at most three numbers.

We study the parameterized complexity of the problem to decide whether a given natural number n satisfies a given $\Delta _0$-formula $\varphi (x)$; the parameter is the size of $\varphi $. This parameterization focusses attention on instances where n is large compared to the size of $\varphi $. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf {AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta _0$ proves the MRDP theorem in a certain weak sense.

This paper proves normalisation theorems for intuitionist and classical negative free logic, without and with the operator for definite descriptions. Rules specific to free logic give rise to new kinds of maximal formulas additional to those familiar from standard intuitionist and classical logic. When is added it must be ensured that reduction procedures involving replacements of parameters by terms do not introduce new maximal formulas of higher degree than the ones removed. The problem is solved by a rule that permits restricting these terms in the rules for $\forall $, $\exists $ and to parameters or constants. A restricted subformula property for deductions in systems without is considered. It is improved upon by an alternative formalisation of free logic building on an idea of Jaśkowski’s. In the classical system the rules for require treatment known from normalisation for classical logic with $\lor $ or $\exists $. The philosophical significance of the results is also indicated.

Continuous functions on the unit interval are relatively tame from the logical and computational point of view. A similar behaviour is exhibited by continuous functions on compact metric spaces equipped with a countable dense subset. It is then a natural question what happens if we omit the latter ‘extra data’, i.e., work with ‘unrepresented’ compact metric spaces. In this paper, we study basic third-order statements about continuous functions on such unrepresented compact metric spaces in Kohlenbach’s higher-order Reverse Mathematics. We establish that some (very specific) statements are classified in the (second-order) Big Five of Reverse Mathematics, while most variations/generalisations are not provable from the latter, and much stronger systems. Thus, continuous functions on unrepresented metric spaces are ‘wild’, though ‘more tame’ than (slightly) discontinuous functions on the reals.

This thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.

For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.

The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.

The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.

The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.

These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.

Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.

Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.

The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.

The abstract was taken directly from the thesis.

Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, Germany

E-mail: pischke@mathematik.tu-darmstadt.de.

URL: https://doi.org/10.26083/tuprints-00026584.

We study the effective versions of several notions related to incompleteness, undecidability, and inseparability along the lines of Pour-El’s insights. Firstly, we strengthen Pour-El’s theorem on the equivalence between effective essential incompleteness and effective inseparability. Secondly, we compare the notions obtained by restricting that of effective essential incompleteness to intensional finite extensions and extensional finite extensions. Finally, we study the combination of effectiveness and hereditariness, and prove an adapted version of Pour-El’s result for this combination.

The one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.

This paper initiates the reverse mathematics of social choice theory, studying Arrow’s impossibility theorem and related results including Fishburn’s possibility theorem and the Kirman–Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in ${\mathsf {RCA}}_0$. We then show that the Kirman–Sondermann analysis of social welfare functions can be carried out in ${\mathsf {RCA}}_0$. This approach yields a proof of Arrow’s theorem in ${\mathsf {RCA}}_0$, and thus in $\mathrm {PRA}$, since Arrow’s theorem can be formalised as a $\Pi ^0_1$ sentence. Finally we show that Fishburn’s possibility theorem for countable societies is equivalent to ${\mathsf {ACA}}_0$ over ${\mathsf {RCA}}_0$.

We prove an isomorphism theorem between the canonical denotation systems for large natural numbers and large countable ordinal numbers, linking two fundamental concepts in Proof Theory. The first one is fast-growing hierarchies. These are sequences of functions on $\mathbb {N}$ obtained through processes such as the ones that yield multiplication from addition, exponentiation from multiplication, etc. and represent the canonical way of speaking about large finite numbers. The second one is ordinal collapsing functions, which represent the best-known method of describing large computable ordinals.

We observe that fast-growing hierarchies can be naturally extended to functors on the categories of natural numbers and of linear orders. The isomorphism theorem asserts that the categorical extensions of binary fast-growing hierarchies to ordinals are isomorphic to denotation systems given by cardinal collapsing functions. As an application of this fact, we obtain a restatement of the subsystem $\Pi ^1_1$-${\mathsf {CA_0}}$ of analysis as a higher-type well-ordering principle asserting that binary fast-growing hierarchies preserve well-foundedness.

Due to Gödel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments—Warren [43] and Murzi and Topey [30]—for the idea that the natural deduction rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s [19] local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e., non-compact. Consequently, I reconsider the use of the $\omega $-rule and I show that the addition of the $\omega $-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement.

Goodstein’s principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments $\mathsf {I}\Sigma _n$ of Peano arithmetic.

A terminating sequent calculus for intuitionistic propositional logic is obtained by modifying the R$\supset $ rule of the labelled sequent calculus $\mathbf {G3I}$. This is done by adding a variant of the principle of a fortiori in the left-hand side of the premiss of the rule. In the resulting calculus, called ${\mathbf {G3I}}_{\mathbf {t}}$, derivability of any given sequent is directly decidable by root-first proof search, without any extra device such as loop-checking. In the negative case, the failed proof search gives a finite countermodel to the sequent on a reflexive, transitive, and Noetherian Kripke frame. As a byproduct, a direct proof of faithfulness of the embedding of intuitionistic logic into Grzegorcyk logic is obtained.