We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $, if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.

Using a variation of Woodin’s $\mathbb {P}_{\mathrm {max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega _{2}$ consisting of functions from $\omega $ to $\omega $. We also show that there can be no such sequence of length $\omega _{4}$.

The article is devoted to comparison of two generalizations of the bounding number $\mathfrak {b}$.

For $\lambda $ inaccessible, we show that “there is a nowhere trivial automorphism of $\mathcal P(\lambda )/[\lambda ]^{<\lambda }$” follows (in ZFC) from $2^\lambda =\lambda ^+$, and is consistent (via forcing) with $2^\lambda>\lambda ^+$.

For every natural number n we introduce a new weak choice principle $\mathrm {nRC_{fin}}$:

Given any infinite set x, there is an infinite subset $y\subseteq x$ and a selection function f that chooses an n-element subset from every finite $z\subseteq y$ containing at least n elements.

By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of $\mathrm {nRC_{fin}}$ to the weak choice principles $\mathrm {RC_m}$ (that has already been studied in [3] and [6]):

Given any infinite set x, there is an infinite subset $y\subseteq x$ with a choice function f on the family of all m-element subsets of y.

Moreover, we prove a stronger analogue of the results in [6] when we study the relation between $\mathrm {nRC_{fin}}$ and $\mathrm {kC_{fin}^-}$ which is defined by

Given any infinite family $\mathcal {F}$ of finite sets of cardinality greater than k, there is an infinite subfamily $\mathcal {A}\subseteq \mathcal {F}$ with a selection function f that chooses a k-element subset from each $A\in \mathcal {A}$.

We prove that the spectrum of Van Douwen families is closed under singular limits. For any maximal eventually different family Raghavan defined in [10] an associated ideal which measures how far the family is from being Van Douwen. Under CH we prove that every ideal containing $\mathrm {Fin}$ is realized as the associated ideal of some maximal eventually different family. Finally, we construct maximal eventually different families with Sacks-indestructible associated ideals to prove that in the iterated Sacks-model every $\aleph _1$-generated ideal containing $\text {Fin}$ is realized.

We present several combinatorial properties of semiselective ideals on the set of natural numbers. The continuum hypothesis implies that the complement of every selective ideal contains a selective ultrafilter, however for semiselective ideals this is not the case. We prove that under certain hypothesis, for instance, $V=L$, there are semiselective ideals whose complement does not contain a selective ultrafilter, and that it is also consistent that the complement of every semiselective ideal contains a selective ultrafilter; specifically, we show that if $V=L$ then there is a generic extension of V where this occurs. We present other results concerning semiselective ideals, namely, an alternative proof of Ellentuck’s theorem for the local Ramsey property, and we prove some facts about the additivity of the ideal of local Ramsey null sets, and also about the generalized Suslin operation on the algebra of local Ramsey sets.

We construct a model in which all C-sequences are trivial, yet there exists a $\kappa $-Souslin tree all of whose limit levels are vanishing levels. This provides an optimal combination of compactness and incompactness. It is obtained by incorporating a so-called mutually exclusive ascent path to Kunen’s 1978 forcing construction, and by analyzing a gallery of $\kappa $-cc forcing extensions of the outcome model.

We show that the statement “In every separable pseudometric space there is a maximal non-strictly $\delta $-separated set.” implies the axiom of choice for countable families of sets. This gives answers to a question of Dybowski and Górka [2]. We also prove several related results.

The development of the forcing method has shown that several key questions regarding infinite sets cannot be settled under ZFC alone. The most widely supported view is that this undecidability simply reflects the limitations of ZFC in addressing all mathematical problems. This perspective has motivated an extensive search for new axioms—the so-called large cardinal axioms—which, when added to ZFC, yield a deeper and more robust understanding of the set-theoretic universe. Along this line, the dissertation is divided into three thematic blocks, each of them framed within these extensions of ZFC:

• • Very large cardinals at the threshold of Kunen inconsistency, with a focus on elementarity and cardinal correctness (Chapter 2).

• • Generalized Descriptive Set Theory at singular strong limit cardinals of countable cofinality, with a focus on two regularity properties (Chapter 3).

• • Covering lemmas and Woodin’s HOD Dichotomy through the lens of Shelah’s pcf theory (Chapter 4).

Specifically, Chapter 2 establishes an inconsistency result using tools from singular cardinal combinatorics and Shelah’s pcf theory, proving the nonexistence of cardinal preserving elementary embeddings into V and establishing thereby a limitation in the hierarchy of large cardinal axioms. The proof is based on the notion of good scales and its connection with Jónsson cardinals. Chapter 3 proves a consistency result obtained via a Prikry-type forcing construction, providing a singular-cardinal analogue of Solovay’s theorem. This thematic block is inspired by Woodin’s Axiom $I_0$ which provides the appropriate axiomatic framework to develop Generalized Descriptive Set Theory in generalized Baire/Cantor spaces at singular cardinals. Concretely, we construct a model of ZFC where $\kappa $ is a strong limit singular cardinal with countable cofinality, and every subset of ${}^\omega \kappa $ in $L(V_{\kappa +1})$ has both the $\kappa $-Perfect Set Property and the $\vec {\mathcal {U}}$-Baire Property. Chapter 4 further explores the study of “covering lemmas” and Woodin’s HOD Dichotomy, employing both the perspective and the tools of pcf theory. Specifically, the connection between the cover property and a new pcf-theoretic concept—called the scale property—is analyzed. This analysis builds a bridge between combinatorial principles in pcf theory and the structural behavior of HOD.

Abstract prepared by Sebastiano Thei.

E-mail: thei91.seba@gmail.com.

URL: https://air.uniud.it/handle/11390/1316344?mode=full.

The Bristol model is an inner model of $L[c]$, where c is a Cohen real, which is not constructible from a set. The idea was developed in 2011 in a workshop taking place in Bristol, but was only written in detail by the author in [8]. This article is a guide for those who want to get a broader view of the construction. We try to provide more intuition that might serve as a jumping board for those interested in this construction and in odd models of $\mathsf {ZF}$. We also correct a few minor issues in the original paper, as well as prove new results. For example, the Boolean Prime Ideal theorem fails in the Bristol model, as some sets cannot be linearly ordered, and the ground model is always definable in its Bristol extensions. In addition to this we include a discussion on Kinna–Wagner Principles, which we think may play an important role in understanding the generic multiverse in $\mathsf {ZF}$.

We introduce finite support iterations of symmetric systems, and use them to provide a strongly modernized proof of David Pincus’ classical result that the axiom of dependent choice is independent over $\operatorname {\mathrm {ZF}}$ with the ordering principle together with a failure of the axiom of choice.

It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V\subseteq M\subseteq N$ of ${\mathsf {ZF}}$ such that N contains no subsets of V that do not already appear in M. We isolate, in the case that M and N are symmetric extensions (particular inner models of a generic extension of V), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity.

We build on a 1990 paper of Bukovský and Copláková-Hartová. First, we remove the hypothesis of ${\mathsf {CH}}$ from one of their minimality results. Then, using a measurable cardinal, we show that there is a $|\aleph _2^V|=\aleph _1$-minimal extension that is not a $|\aleph _3^V|=\aleph _1$-extension, answering the first of their questions.

For any $2 \le n < \omega $, we introduce a forcing poset using generalized promises which adds a normal n-splitting subtree to a $(\ge \! n)$-splitting normal Aronszajn tree. Using this forcing poset, we prove several consistency results concerning finitely splitting subtrees of Aronszajn trees. For example, it is consistent that there exists an infinitely splitting Suslin tree whose topological square is not Lindelöf, which solves an open problem due to Marun. For any $2 < n < \omega $, it is consistent that every $(\ge \! n)$-splitting normal Aronszajn tree contains a normal n-splitting subtree, but there exists a normal infinitely splitting Aronszajn tree which contains no $(< \! n)$-splitting subtree. To show the latter consistency result, we prove a forcing iteration preservation theorem related to not adding new small-splitting subtrees of Aronszajn trees.

For cardinals $\mathfrak {a}$ and $\mathfrak {b}$, we write $\mathfrak {a}=^\ast \mathfrak {b}$ if there are sets A and B of cardinalities $\mathfrak {a}$ and $\mathfrak {b}$, respectively, such that there are partial surjections from A onto B and from B onto A. $=^\ast $-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf {ZF}+\mathsf {DC}_\kappa $, where $\kappa $ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165–207 (1984)]. Nevertheless, we show that surjective cardinals form a “surjective cardinal algebra”, whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot \mathfrak {a}=^\ast m\cdot \mathfrak {b}$ implies $\mathfrak {a}=^\ast \mathfrak {b}$ for all cardinals $\mathfrak {a},\mathfrak {b}$ and all nonzero natural numbers m.

We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak {i}<\mathfrak {s}_{1/2}$), Con($\mathfrak {r}_{1/2}<\mathfrak {b}$), and Con($\mathfrak {i}_*<2^{\aleph _0}$). This answers two questions raised in [5]. Besides, we prove the consistency of $\mathfrak {s}_{1/2}^{\infty } < $ non$(\mathcal {E})$ and cov$(\mathcal {E}) < \mathfrak {r}_{1/2}^{\infty }$, where $\mathcal {E}$ is the $\sigma $-ideal generated by closed sets of measure zero.

We prove that P-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph _{\omega } $-many random reals to a model of CH.These results improve and correct previous theorems that can be found in the literature.

In this work, we consider the ideals $m^0(\mathcal {I})$ and $\ell ^0(\mathcal {I})$, ideals generated by the $\mathcal {I}$-positive Miller trees and $\mathcal {I}$-positive Laver trees, respectively. We investigate in which cases these ideals have cofinality larger than $\mathfrak {c}$ and we calculate some cardinal invariants closely related to these ideals.

We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa $-tree $\mathbf T$. This is an invariant in the sense that if $\mathbf T$ and $\mathbf T'$ are club-isomorphic, then $V(\mathbf T)\mathbin {\bigtriangleup } V(\mathbf T')$ is nonstationary. Additional features of this invariant imply that the spectrum is closed under finite unions and intersections. The set $V(\mathbf T)$ must be stationary for a homogeneous normal $\kappa $-Aronszajn tree $\mathbf T$, and if there exists a special $\kappa $-Aronszajn tree, then there exists one $\mathbf T$ that is homogeneous and satisfies that $V(\mathbf T)$ covers a club in $\kappa $. It is consistent (from large cardinals) that there is an $\aleph _2$-Souslin tree, and yet $V(\mathbf T)$ is co-stationary for every $\aleph _2$-tree $\mathbf T$. Both $V(\mathbf T)=\emptyset $ and $V(\mathbf T)=\kappa $ (modulo nonstationary) are shown to be feasible using $\kappa $-Souslin trees, even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^\kappa $ many $\kappa $-Souslin trees for which the corresponding family of vanishing levels forms an antichain in the Boolean algebra of the powerset of $\kappa $, modulo the nonstationary ideal.