Related papers: Set Theory in the Foundation of Math; Internal Cla…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
This paper introduces a new theory which encompasses concepts and ideas from set theory, type theory, and Le\'{s}niewski's mereology and describes its possibility as an alternative foundation for mathematics. In the introduction section I…
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…
We study elementary theories of well-pointed toposes and pretoposes, regarded as category-theoretic or "structural" set theories in the spirit of Lawvere's "Elementary Theory of the Category of Sets". We consider weak intuitionistic and…
Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows…
Isomorphism is central to the structure of mathematics and has been formalized in various ways within dependent type theory. All previous treatments have done this by replacing quantification over sets with quantification over groupoids of…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
We present a first-order theory of sequences with integer elements, Presburger arithmetic, and regular constraints, which can model significant properties of data structures such as arrays and lists. We give a decision procedure for the…
We propose a set theory strong enough to interpret powerful type theories underlying proof assistants such as LEGO and also possibly Coq, which at the same time enables program extraction from its constructive proofs. For this purpose, we…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are…
As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise…
We deal with stability theory for ``reasonable'' non-elementary classes without any remanents of compactness (like: above Hanf number or definable by L_{omega_1, omega}).
The basics of set theory are usually copied, directly or indirectly, by computer scientists from introductions to mathematical texts. Often mathematicians are content with special cases when the general case is of no mathematical interest.…
A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete…
We propose a natural theory SO axiomatizing the class of sets of ordinals in a model of ZFC set theory. Both theories possess equal logical strength. Constructibility theory in SO corresponds to a natural recursion theory on ordinals.
What makes sets, or more precisely, the category {\bf Set} important in Mathematics are the well known {\it two} specific ways in which arbitrary mappings $f : X \longrightarrow Y$ between any two sets $X, Y$ can {\it fail} to be…
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a…
A selection of the relevant theorems of Probability Theory that comes directly from Kolmogorov's axioms, Set Theory basic results, definitions and rules of inference are listed and proven in a systematic approach, aiming the student who…