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)

It is proved in $\mathsf {ZF}$ (without the axiom of choice) that, for all infinite sets M, there are no surjections from $\omega \times M$ onto $\operatorname {\mathrm {\mathscr {P}}}(M)$.

For a set M, let |M| denote the cardinality of M. A family F is called strongly almost disjoint if there is an n∈ω such that |A∩B|<n for any two distinct elements A, B of F. It is shown in ZF (without the axiom of choice) that, for all infinite sets M and all strongly almost disjoint families F⊆P(M), |F|<|P(M)| and there are no finite-to-one functions from P(M) into F, where P(M) denotes the power set of M.

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)

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)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

We write$\mathcal {S}_n(A)$for the set of permutations of a setAwithnnon-fixed points and$\mathrm {{seq}}^{1-1}_n(A)$for the set of one-to-one sequences of elements ofAwith lengthnwherenis 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 setsA. Among our results, we show, in ZF, that$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$for any infinite setAif${\mathrm {AC}}_{\leq n}$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 setAand the subscript$n+1$cannot be reduced ton. Moreover, (...) we also show that “$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$for any infinite setA” 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)

We write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality, where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals. We show, in ZF, that if is assumed, then for any infinite cardinal. Moreover, the assumption cannot be removed for and the superscript cannot be replaced by n. We (...) also show under that for any infinite cardinal, implies is Dedekind‐infinite. (shrink)

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)

A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$. On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following: (1) If there is a finitary (...) partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A. (2) For all $n\in \omega $, $|A^n|<|\mathscr {B}(A)|$. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A. (shrink)

We write and for the cardinalities of the set of finite subsets and the set of permutations with finitely many non‐fixed points, respectively, of a set which is of cardinality. In this paper, we investigate relationships between and for an infinite cardinal in the absence of the Axiom of Choice. We give conditions that make and comparable as well as give related consistency results.

We show that (i) it is consistent with that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind‐infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of ; and (iv) an infinite set X is weakly (...) Dedekind‐infinite if and only if has infinite towers if and only if X has countable partitions. (shrink)