An axiomatic account of multiset theory is given, where multiplicities are of the same sort as sets (with their own internal structures). Various theories are proposed covering different existing multiset systems, as well as a stronger theory which is equiconsistent with Zermelo–Fraenkel set theory and with antifoundation. The inclusion relation receives a recursive definition in terms of membership and is shown to be not always antisymmetric.

We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{\ast }$ and $A-{\mathsf{BMM}}^{\ast ,++}$ (where $A$ is a universally Baire set of reals), and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{\ast ,++}$ holds true for every set of reals $A$ in $L\left(\mathbb{R}\right)$, then Woodin’s axiom $(\ast )$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{\ast ,++}$ for every $A$ which is universally Baire.

We characterize the situation of having at least $(2^{\kappa }{)}^{+}$-many normal ultrafilters on a measurable cardinal $\kappa$. We also show that if $\kappa$ is a compact cardinal, then $\kappa$ carries $(2^{\kappa }{)}^{+}$-many $\kappa$-complete ultrafilters, each of which extends the club filter on $\kappa$.

The transitivity of the relation of part to whole is often questioned. But it is among the most basic principles of mereology. In this paper we present a general solution to the problem of transitivity of parthood which may be satisfactory for both its advocates and its opponents.

We will show that even without the transitivity of parthood one can define—basic in mereology—the notion of being a mereological sum of some objects. We formulate several proposals of general approaches to the concept of being a part of a whole, none of which contains any existential assumptions. By adding the transitivity of parthood we obtain an axiomatization of “existentially neutral” (or “nonexistential”) mereology.

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.

A paper by Beneš, published in 1954, was an attempt to prove the consistency of $\mathsf{NF}$ (Quine’s set theory “New Foundations”) via a partial model of Hailperin’s finite axiomatization of $\mathsf{NF}$. Here, I offer an analysis of Beneš’s proof in a De Giorgi-style setting for set theory. This approach leads to an abstract version of Beneš’s theorem that emphasizes the monotone and invariant content of the axioms proved to be consistent, in a sense of monotony and invariance that this paper intends to state rigorously and to help clarify. Moreover, some tentative speculation will be made about possible developments of the topic in the following two directions: (1) which set theories can be proved to be consistent via Beneš-like constructions, and (2) how can we elaborate on Beneš’s model to get a consistency proof for full $\mathsf{NF}$?

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 ${dfin}^{\ast }\left(M\right)$ denote the set of all weakly Dedekind finite subsets of $M$. In this paper, we prove, in Zermelo–Fraenkel (ZF) set theory, that $\left|{dfin}^{\ast }\right(M\left)\right|<\left|\mathcal{P}\right(M\left)\right|$ if ${dfin}^{\ast }\left(M\right)$ is Dedekind infinite, whereas $\left|{dfin}^{\ast }\right(M\left)\right|<\left|\mathcal{P}\right(M\left)\right|$ cannot be proved from ZF for an arbitrary $M$.

We prove that in ${NTP}_2$ theories the dp-rank of a type can be witnessed by indiscernible sequences of tuples satisfying that type. If the type has dp-rank infinity, then this can be witnessed by singletons (in any theory).