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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…
One measure of the complexity of a first-order theory, and similarly a type, is the complexity of the formulas required to axiomatize it. We say a theory is bounded if there is an axiomatization involving only $\forall_n$-formulas for some…
This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative…
Mathematical theories are classified in two distinct classes : {\it rigid}, and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory, topology, category theory, etc., have a basic concept - given for instance by a set…
This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J_2 (the second set in Jensen's constructible hierarchy). The implication, well-known to…
A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…
Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…
Locales have been studied as "topologies without points", mainly by tools of category theory. While traditional topology presents a space as a set of points with specified neighborhoods, localic topology presents a space as a lattice of…
The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…
The paper is a naive introduction to descriptive set theory. It is aimed mathematicians without a background in logic. The goal is to provide the basic facts used for applications of descriptive set theory to other areas of mathematics,…
The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…
It is well known that the R, the set of real numbers, is an abstract set, where almost all its elements cannot be described in any finite language. We investigate possible approaches to what might be called an epi-constructionist approach…
This paper describes the homology of various simplicial complexes associated to set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition…
We develop a domain-theoretic framework for imprecise probability reasoning and inference on general topological spaces with a countably based continuous lattice of open sets. We address two distinct forms of uncertainty: partial or…
We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower…
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much…
This is the second in a series of papers on the relation between algebraic set theory and predicative formal systems. In part I, we introduced the notion of a predicative category of small maps and obtained the result that such categories…
The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…