Two consistency results on set mappings

1998

It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.

Notre Dame Journal of Formal Logic, 2018

In this paper we study set mappings on 4-tuples. We continue a previous work of Komjath and Shelah by getting new finite bounds on the size of free sets in a generic extension. This is obtained by an entirely different forcing construction. Moreover we prove a ZFC result for set mappings on 4-tuples and also as another application of our forcing construction we give a consistency result for set mappings on triples.

Notre Dame Journal of Formal Logic, 1989

HajnaPs free set principle is equivalent to the axiom of choice, and some of its variants for Dedekind-finite sets are equivalent to countable forms of the axiom of choice.

Notes on Number Theory and Discrete Mathematics, 2024

This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, 'm-cardinality', is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, 'm-cardinal numbers' are defined as a generalization of cardinal numbers in the context of multisets. Some properties of m-cardinal numbers associated with finite and infinite msets have been researched. Concrete examples of transfinite m-cardinal numbers are given, corresponding to infinite msets which are less than ℵ 0 (the cardinality of the countably infinite set). It has been established that between finite numbers and ℵ 0 there exist hierarchies of transfinite m-cardinals, corresponding to infinite msets. Furthermore, there are examples of infinite msets with negative multiplicity that have a cardinality less than zero. We prove that there is a decreasing sequence of transfinite m-cardinal numbers, corresponding to infinite msets with negative multiplicity, and in this sequence, there is not a smallest transfinite m-cardinal number.

Research Gate, 2024

In an earlier paper concerning n:m correspondence of infinite sets, 1 the correspondence (mapping) was shown in a tabular manner (starting from the same domain). Here we consider using functions to do the mapping. When using tabular mapping, it is easy to consider both sets as being picked from the same domain. With functional mapping it is easy to make mistakes between the required domain, codomain, and range of the functions. In the first instance, functional mapping is therefore ill advised. New set properties define the upper bounds of ℕ, ℤ, ℚ, and ℝ so that tabular mapping between dissimilar domains is numerically consistent.

International Journal of Open Problems in Computer Science and Mathematics, 2012

The author introduces the concept of intrinsic set property, by means of which the well-known Cantor's Theorem can be deduced. As a natural consequence of this fact, it is proved that Cantor's Theorem need not imply the existence of a tower of different-size infinities, because the impossibility of defining a bijection between any infinite countable set and its power can be a consequence of the existence of any intrinsic property which does not depend on size.

Russian Mathematical Surveys, 2016

2009

Transactions of the American Mathematical Society, 1976

We construct a permutation model of set theory with urelements in which C2 (the choice principle restricted to families whose elements are unordered pairs) is false but the principle, "For every infinite cardinal m, 2m = m" is true. This answers in the negative a question of Tarski posed in 1924. We note in passing that the choice principle restricted to well-ordered families of finite sets is also true in the model.

We give several partial positive answers to a question of Juhász and Szentmiklóssy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.