Suppose I have a collection of "elements" together with operations that satisfy the axioms for a commutative ring with identity --- except that these elements form not a set, but a …

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spa …

Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular …

The power set of every infinite set is uncountable. An infinite set (as an element of the power set) cannot be defined by writing the infinite sequence of its elements but only by …

Many years ago, Takeuti constructed a first order theory of ordinal numbers which we shall denote by ORD(T) and proved that ZF+(V=L) could be interpreted in ORD(T). Let F(x) design …

The Galvin-Prikry theorem says that Borel sets are Ramsey. This means that for every Borel set $S\subseteq[\omega]^\omega$, there is an $A\in[\omega]^\omega$ such that either $[A]^ …

As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is …

Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$. Given $x …

Suppose we have a set $M = (0,1) \subset R$ of reals well-ordered as the first uncountable ordinal. Let $M(a) = \lbrace x \in M : x < a \rbrace$. For every $a \in M$ set $M(a)$ …

In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and Z …

Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis. What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ …

[Apologies in advance for a fluffy question] I'm reading this old paper by Pelletier, where he gives a Boolean-valued model version of class forcing, assuming that the Boolean alg …

In Jech's SET THEORY (a very early edition to which I have access), it is shown that the existence of 0-sharp implies the existence of a truth definition for the constructible univ …

Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig ident …