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. (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. (2) For all $n\in \omega $, $|A^n|<|\mathscr {B}(A)|$.

3. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

The present note was prompted by Weber’s approach to proving Cantor’s theorem, i.e., the claim that the cardinality of the power set of a set is always greater than that of the set itself. While I do not contest that his proof succeeds, my point is that he neglects the possibility that by similar methods it can be shown also that no non-empty set satisfies Cantor’s theorem. In this paper unrestricted abstraction based on a cut free Gentzen type sequential calculus will be employed to prove both results. In view of the connection between Priest’s three-valued logic of paradox and cut free Gentzen calculi this, a fortiori, has an impact on any paraconsistent set theory built on Priest’s logic of paradox.

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 $\operatorname {\mathrm {\mathscr {P}}}(M)$.

The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof.