Related papers: Totalities of Infinite Sets
In this Phd. thesis, a structural analysis of construction schemes is developed. The importance of this study will be justified by constructing several distinct combinatorial objects which have been of great interest in mathematics. We then…
Three types of Cantor sets are studied.For any integer $m\ge 4$, we show that every real number in $[0,k]$ is the sum of at most $k$ $m$-th powers of elements in the Cantor ternary set $C$ for some positive integer $k$, and the smallest…
When a proposition has no proof in an inference system, it is sometimes useful to build a counter-proof explaining, step by step, the reason of this non-provability. In general, this counter-proof is a (possibly) infinite co-inductive proof…
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three…
We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa+1})$ has both the $\kappa$-Perfect Set Property and the $\mathcal{\vec{U}}$-Baire Property. This is a higher analogue of…
It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…
This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
The first part of this article deals with theorems on uniqueness in law for \sigma-finite and constructive countable random sets, which in contrast to the usual assumptions may have points of accumulation. We discuss and compare two…
In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions…
New measures for the quantization of systems with constraints are discussed and applied to several examples, in particular, examples of alternative but equivalent formulations of given first-class constraints, as well as a comparison of…
We investigate a notion called uniqueness in power kappa that is akin to categoricity in power kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite…
On a finite structure, the polymorphism invariant relations are exactly the primitively positively definable relations. On infinite structures, these two sets of relations are different in general. Infinitary primitively positively…
The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole…
To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…
According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set…
We construct a combinatorially large measure zero subset of the Cantor set.
Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of…
We provide a constructive algorithm to find the best separable approximation to an arbitrary density matrix of a composite quantum system of finite dimensions. The method leads to a condition of separability and to a measure of…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…