Related papers: Four Cardinals and Their Relations in ZF

In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite…

For a set $x$, let $\mathcal{S}(x)$ be the set of all permutations of $x$. We study several aspects of this notion in $\mathsf{ZF}$. The main results are as follows: (1) $\mathsf{ZF}$ proves that for all sets $x$, if $\mathcal{S}(x)$ is…

This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that…

A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}_{n}(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set…

A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…

Let $X$ be a compact metric space and let $|A|$ denote the cardinality of a set $A$. We prove that if $f\colon X\to X$ is a homeomorphism and $|X|=\infty$ then for all $\delta>0$ there is $A\subset X$ such that $|A|=4$ and for all $k\in Z$…

For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite…

Given a nonempty finite multiset $S$ of positive integers, we wish to find a partially ordered set $P$ of minimal cardinality such that the multiset of cardinalities of all maximal chains in $P$ equals $S$. This paper establishes upper and…

Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function…

Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a…

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean…

Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{\alpha}_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$…

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper…

We compute the cardinality $\mathfrak n_{\dim}(\mathcal M)$ of the sets of dimension functions on the ordered structures $\mathcal M$. The inequality $\mathfrak n_{\dim}(\mathcal M) \leq 1$ holds if $\mathcal M$ is a d-minimal expansion of…

A system $\mathcal M$ of equivalence relations on a set $E$ is \emph{semirigid} if only the identity and constant functions preserve all members of $\mathcal M$. We construct semirigid systems of three equivalence relations. Our…

A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of…

We prove, via transfinite recursion, the existence, inside any linearly ordered set of appropriate regular cardinality $\lambda$, of a particular kind of well-ordered subsets characterized by the property of $\lambda$-fullness. Let $H$ be a…

Given a set $X$ of $n\times n$ matrices and a positive integer $m$, we consider the problem of estimating the cardinalities of the product sets $A_1 \dotsc A_m$, where $A_i\in X$. When $X=\mathcal M_n(\mathbb{Z};H)$, the set of $n\times n$…

Fairly deep results of Zermelo-Frenkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is K*K = K,…

We prove that any definable family of subsets of a definable infinite set $A$ in an o-minimal structure has cardinality at most $|A|$. We derive some consequences in terms of counting definable types and existence of definable topological…