Related papers: Intermediate models with deep failure of choice
Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $\mathsf{KWP}$ to denote that there is some $\alpha$ for which this holds. The Kinna--Wagner Conjecture, formulated…
We construct a model $M$ of ZF which lies between $L$ and $L[c]$ for a Cohen real $c$ and does not have the form $L(x)$ for any set $x$. This is loosely based on the unwritten work done in a Bristol workshop about Woodin's HOD Conjecture in…
We continue the work done by Gitik, Kanovei, Koepke, and later by the authors. We prove that for every set $A$ in a Magidor-Radin generic extension using a coherent sequence such that $o^{\vec{U}}(\kappa)<\kappa$, there is a subset $C'$ of…
The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique…
The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of ZF between the ground model and the generic extension, and often the axiom of choice…
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…
We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\neg\square_{\omega_3}+\neg\square(\omega_3)$. Then,…
A $\Sigma$-construction of Solovay is extended to the case of intermediate sets which are not necessarily subsets of the ground model, with a more transparent description of the resulting forcing notion than in the classical paper of…
According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $\lambda$ and non-trivial elementary embedding $j:V_{\lambda+2}\to V_{\lambda+2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone has been…
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_\delta$ be a non-trivial $\in$-cofinal $\Sigma_1$-elementary embedding, where $\varepsilon,\delta$ are limit ordinals. We prove some restrictions on the constructibility of…
In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly…
The theory ZFC implies the scheme that for every cardinal $\delta$ we can make $\delta$ many dependent choices over any definable relation without terminal nodes. Friedman, the first author, and Kanovei constructed a model of ZFC$^-$ (ZFC…
In "Extensional realizability for intuitionistic set theory", we introduced an extensional variant of generic realizability, where realizers act extensionally on realizers, and showed that this form of realizability provides "inner" models…
In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving…
We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a…
A subset of a model of ${\sf PA}$ is called neutral if it does not change the $\mathrm{dcl}$ relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in…
In this paper, we build Fidel-structures valued models following the methodology developed for Heyting-valued models; recall that Fidel structures are not algebras in the universal algebra sense. Taking models that verify Leibniz law, we…
We show that if $M$ is a countable transitive model of ZF and if $a,b$ are reals not in $M$, then there is a $G$ generic over $M$ such that $b \in L[a,G]$. We then present several applications such as the following: if $J$ is any countable…
For a nondegenerate additive subgroup $G$ of the $n$-dimensional vector space $F^n$ over an algebraically closed field $F$ of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type $W(G,n)$ spanned by all…
A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function $f$ defined on the positive integers and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the partial…