Related papers: Well ordering principles for iterated $\Pi^1_1$-co…

Properties expressed as the provability of a first-order sentence can be disproved by just finding a model of the negation of the sentence. This fact, however, is meaningful in restricted cases only, depending on the shape of the sentence…

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic $L_{2}$. A major theme in RM is therefore the study…

We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results…

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…

It is a striking fact from reverse mathematics that almost all theorems of countable and countably representable mathematics are equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of…

In this paper we calibrate the strength of the soundness of a Kripke-Platek set theory with the axioms of Infinity and \Pi_{1}-Collection with the assumption that`there exists an uncountable regular ordinal' in terms of the existence of…

In the lecture notes it is shown that an ordinal $\psi_{\Omega}(\varepsilon_{\mathbb{S}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}\omega+(M\prec_{\Sigma_{1}}V)$. In this note we show that ${\sf…

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of…

Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all…

This paper presents a conformal prediction method for classification in highly imbalanced and open-set settings, where there are many possible classes and not all may be represented in the data. Existing approaches require a finite, known…

The set theory KP$\Pi_{N+1}$ for $\Pi_{N+1}$-reflecting universes is shown to be $\Pi_{N+1}$-conservative over iterations of $\Pi_{N}$-recursively Mahlo operations for each $N\geq 2$.

The topic of comprehensibility of machine-learned theories has recently drawn increasing attention. Inductive Logic Programming (ILP) uses logic programming to derive logic theories from small data based on abduction and induction…

We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for…

In classical set theory, the ordinals form a linear chain that we often think of as a very thin portion of the set-theoretic universe. In intuitionistic set theory, however, this is not the case and there can be incomparable ordinals. In…

Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually…

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained…

We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible…

A key feature of inductive logic programming (ILP) is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs.…

We introduce and study several notions of computability-theoretic reducibility between subsets of $\omega$ that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be…