Related papers: Families of Sets in Bishop Set Theory
We apply fundamental notions of Bishop set theory (BST), an informal theory that complements Bishop's theory of sets, to the theory of Bishop spaces, a function-theoretic approach to constructive topology. Within BST we develop the notions…
Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We…
We present the first steps of a predicative reconstruction of the constructive Bishop-Cheng measure theory. Working in a semi-formal elaboration of Bishop's set theory and invoking the notion of a set-indexed family of subsets (of a given…
Bishop's informal set theory is briefly discussed and compared to Lawvere's Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive…
We study a new model theory for formal mathematical systems that we developed in a previous paper. We introduce isomorphic and homomorphic structures for formal languages, present some results and examples and conclude our paper with a…
In this article we survey some of the recent developments in the structure theory of set addition.
We introduce the notion of a Bishop topological group i.e., a group X equipped with a Bishop topology of functions F such that the group operations of X are Bishop morphisms with respect to F. A closed subset in the neighborhood structure…
We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We…
We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.
We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…
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 develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more…
In this paper we give an ordinal analysis of a set theory with $\Pi_{1}$-Collection.
In this paper we give an ordinal analysis of a set theory with $\Pi_{N}$-Collection.
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected…
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. Starting from ZFC, the exposition in this first part includes relation and order theory as well as a construction of…
Using ideas from synthetic topology, a new approach to descriptive set theory is suggested. Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.…
We present an abstract framework for asymptotic analysis of convergence based on the notions of eventual families of sets that we define. A family of subsets of a given set is called here an "eventual family" if it is upper hereditary with…
We pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to…