We write for the cardinality of the set of finite sequences of a set which is of cardinality. With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality. In this paper, we show that “ for every cardinal ” is provable in and this is the best possible result in the absence of. Similar results are also obtained for : the cardinality of the set of finite (...) sequences without repetition of a set which is of cardinality. (shrink)

In this article, I develop a novel version of the multiverse theory of sets called hierarchical pluralism by introducing the notion of “degrees of intentionality” of theories. The presented view is articulated for the purpose of reconciling epistemological realism and the multiverse theory of sets so as to preserve a considerable amount of epistemic objectivity when working with the multiverse theory. I give some arguments in favor of a hierarchical picture of the multiverse in which theories or models are thought (...) to be ordered with respect to their plausibility as a manifestation of endorsing the idea that some set theories are more plausible than others. The proposed multiverse account settles the pluralist’s dilemma, the dichotomy that there is a trade-off between the richness of mathematical ontology and the objectivity of mathematical truth. The view also extends and serves as an alternative position to Balaguer’s intention-based Platonism, from which he claims that a certain version of mathematical pluralism follows. (shrink)

In the Zermelo–Fraenkel set theory (ZF), |$|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$| for any infinite set |$A$|⁠, where |$\textrm {fin}(A)$| is the set of finite subsets of |$A$|⁠, |$2^{|A|}$| is the cardinality of the power set of |$A$| and |$\textrm {Part}(A)$| is the set of partitions of |$A$|⁠. In this paper, we show in ZF that |$|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$| for any set |$A$| with |$|A|\geq 5$|⁠, where |$\textrm {Part}_{\textrm {fin}}(A)$| is the set of partitions of |$A$| whose members are finite. We (...) also show that, without the Axiom of Choice, any relationship between |$|\textrm {Part}_{\textrm {fin}}(A)|$| and |$2^{|A|}$| for an arbitrary infinite set |$A$| cannot be concluded. (shrink)

We write and for the cardinalities of the set of finite sequences and the set of finite subsets, respectively, of a set which is of cardinality. With the axiom of choice (), for every infinite cardinal but, without, any relationship between and for an arbitrary infinite cardinal cannot be proved. In this paper, we give conditions that make and comparable for an infinite cardinal. Among our results, we show that, if we assume the axiom of choice for sets of finite (...) sets, then for every Dedekind‐infinite cardinal and the condition that is Dedekind‐infinite cannot be weakened to weakly Dedekind‐infinite. (shrink)

A set x is Dedekind infinite if there is an injection from ω into x; otherwise x is Dedekind finite. A set x is power Dedekind infinite if math formula, the power set of x, is Dedekind infinite; otherwise x is power Dedekind finite. For a set x, let pdfin be the set of all power Dedekind finite subsets of x. In this paper, we prove in math formula two generalizations of Cantor's theorem : The first one is that for (...) all power Dedekind infinite sets x, there are no Dedekind finite to one maps from math formula into pdfin. The second one is that for all sets math formula, if x is infinite and there is a power Dedekind finite to one map from y into x, then there are no surjections from y onto math formula. We also obtain some related results. (shrink)

This article proposes a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor’s account if the axiom of choice is true, but its consequences differ from those of Cantor’s if the axiom of choice is false. This article argues that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set (...) that is bigger than it—that can arise from Cantor’s account if the axiom of choice is false, illuminates the debate over whether the axiom of choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

For a cardinal, we write for the cardinality of the set of permutations with finitely many non‐fixed points of a set which is of cardinality. We investigate the relationships between and for an arbitrary infinite cardinal in (without the axiom of choice). It is proved in that for all infinite cardinals, and we show that this is the best possible result.

For a set M, let \\) denote the set of all finite sequences which can be formed with elements of M, and let \ denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \: There exists a set M such that Open image in new window and no function Open image (...) in new window is finite-to-one. (shrink)

We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$. With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm {{seq}}^{1-1}_n(A)|$ are equal for all infinite sets A. Among our results, we show, in ZF, that $|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$ for any infinite set A if ${\mathrm {AC}}_{\leq n}$ is (...) assumed and this assumption cannot be removed. In the other direction, we show that $|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A and the subscript $n+1$ cannot be reduced to n. Moreover, we also show that “ $|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A” is not provable in ZF. (shrink)

A set $A$ is Dedekind infinite if there is a one-to-one function from $\omega$ into $A$. A set $A$ is weakly Dedekind infinite if there is a function from $A$ onto $\omega$; otherwise $A$ is weakly Dedekind finite. For a set $M$, let $\operatorname{dfin}^{*}$ denote the set of all weakly Dedekind finite subsets of $M$. In this paper, we prove, in Zermelo–Fraenkel set theory, that $|\operatorname{dfin}^{*}|\lt |\mathcal{P}|$ if $\operatorname{dfin}^{*}$ is Dedekind infinite, whereas $|\operatorname{dfin}^{*}|\lt |\mathcal{P}|$ cannot be proved from ZF for (...) an arbitrary $M$. (shrink)