Mathematics

We force over the constructible universe to obtain a model of the Π 1 3 -reduction property, thus lowering the best known large cardinal strength from Δ 1 2 -determinacy to just ZFC.

Let A i :=( A i , ≤ i , ⊤ i , ⊙ i , → i , ⊥ i ) , i=1,2 be two complete residuated multilattices, G (set of objects) and M (set of attributes) be two nonempty sets and (φ,ψ) a Galois connection between A G 1 and A M 2 . In this work we prove that C:={(h,f)∈ A G 1 × A M 2 ∣φ(h)=f and ψ(f)=h} is a complete residuated multilattice. This is a generalization of a result by Ruiz-Calvi{ñ}o and Medina \cite{RM12} saying that if the (reduct of the) algebras A i , i=1,2 are complete multilattices, then C is a complete multilattice.

We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating the recursively enumerable relations between lists of terms, the basic objects under consideration. A recursive system consists of axioms, which are special quantifier-free positive horn formulas, and of specific rules of inference. Its extension to formal mathematical systems leads to a formal structural induction with respect to the axioms of the underlying recursive system. This approach provides some new representation theorems without using artificial and difficult interpretation techniques. Within this frame we will also derive versions of Gödel's First and Second Incompleteness Theorems for a general class of axiomatized formal mathematical systems.

In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and simple enough to use in formal proofs. Based on this definition, we formalize the concepts of subgroup, coset, conjugate, normal subgroup, and quotient group, and formally derive some related theorems. We aim to keep these formalizations transparent and concise, and as close as possible to the standard mathematical theory. The results can be implemented in proof assistants that are based on calculus of constructions.

Type theory plays an important role in foundations of mathematics as a framework for formalizing mathematics and a base for proof assistants providing semi-automatic proof checking and construction. Derivation of each theorem in type theory results in a formal term encapsulating the whole proof process. In this paper we use a variant of type theory, namely the Calculus of Constructions with Definitions, to formalize the standard theory of binary relations. This includes basic operations on relations, criteria for special properties of relations, invariance of these properties under the basic operations, equivalence relation, well-ordering, and transfinite induction. Definitions and proofs are presented as flag-style derivations.

We introduce a new foundation rank based in the relation of dividing between partial types. We call DU to this rank. We also introduce a new way to define the D rank over formulas as a foundation rank. In this way, SU, DU and D are foundation ranks based in the relation of dividing. We study the properties and the relations between these ranks. Next, we discuss the possible definitions of a supersimple type. This is a concept that it is not clear in the previous literature. In this paper we give solid arguments to set up a concrete definition of this concept and its properties. We also see that DU characterizes supersimplicity, while D not.

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. Thus we will begin a view that online algorithms can be viewed as a sub-area of computable analysis. We will give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. Conversely, we also get an enrichment of computable analysis from ideas from the analysis of classical online algorithms.

Frege's definition of the real numbers, as envisaged in the second volume of \textit{Grundgesetze der Arithmetik}, is fatally flawed by the inconsistency of Frege's ill-fated \textit{Basic Law V}. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.

Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt Gödel (1932), and it is proved by Stanisław Jaśkowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). Gödel's proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Gödel or the Gödel-Dummet Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definablility of propositional connectives in this logic.

Harvey Friedman's gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: it arises via iterated applications of a uniform Kruskal theorem.