3
votes
1answer
59 views
Proper-class sized “ring” with no maximal ideals
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 …
4
votes
2answers
222 views
How do we express measurable spaces using type theory?
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 …
6
votes
0answers
66 views
Existence of a regular subposet which collapses everything except the top cardinal
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 …
9
votes
3answers
912 views
What would remain of current mathematics without axiom of power set? [closed]
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 …
3
votes
0answers
140 views
A question about interpreting set theories containing large cardinal axioms in theories of ordinal numbers that are extensions of arithmetic
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 …
10
votes
1answer
372 views
Why does the generalised Galvin-Prikry Theorem only hold at Ramsey cardinals?
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]^ …
7
votes
1answer
186 views
Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game
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 …
3
votes
1answer
178 views
Trivial forcings which are not very trivial
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 …
1
vote
4answers
340 views
The paradox with the first uncountable ordinal
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 $M(a)$ …
1
vote
0answers
70 views
A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
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 …
0
votes
2answers
415 views
Ontological status of some “sets” in ZFC [closed]
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{ …
6
votes
3answers
230 views
Class forcing: Pelletier vs Friedman
[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 …
5
votes
1answer
84 views
A Question Regarding the Relation Between 0-sharp and Koepke’s Bounded Truth Predicate.
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 …
12
votes
3answers
339 views
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 construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?
10
votes
2answers
321 views
Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?
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 …

