This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...

learn more… | top users | synonyms

11
votes
0answers
163 views

How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is ...
11
votes
0answers
401 views

Formulations of Singular Cardinals Hypothesis

I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are: SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
10
votes
0answers
133 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
10
votes
0answers
149 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
10
votes
0answers
181 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
9
votes
0answers
115 views

Does such a first-order theory exist? A question pertaining to transitive models of ZFC.

Assume a proper class of inaccessibles. Does there exist a first-order theory $T$ subject to the following conditions? $T$ admits an infinite model Whenever $M$ is a transitive model of ZFC with $T ...
9
votes
0answers
83 views

Choice in a Particular HOD Type Model

Let $V \models ZFC$. Let $P$ be a forcing and $G \subseteq P$ be generic over $V$. Let $x \in V[G]$. Let $M$ be the class of set that are hereditarily definable (in $V[G]$) using as parameters $x$ ...
8
votes
0answers
215 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...
7
votes
0answers
94 views

Carlson's model and Sierpinski sets

Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure (This is true in Carlson's model which is obtained by ...
7
votes
0answers
269 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
7
votes
0answers
175 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
7
votes
0answers
328 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
6
votes
0answers
188 views

Is Kunen's claim about non-equivalent forms of Axiom of Choice, true?

Consider the following forms of the axiom of choice: $AC_1:\forall F\neq \emptyset~~~(\emptyset\notin F~\wedge~\forall x,y\in F~~~(x\neq y\rightarrow x\cap y= \emptyset))\rightarrow \exists C~\forall ...
6
votes
0answers
80 views

Connectedness of parts used in the Banach–Tarski paradox

A quote from the Wikipedia article "Axiom of choice": One example is the Banach–Tarski paradox which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many ...
6
votes
0answers
128 views

Skolem Hulls in $H_{\omega_2}$

Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \epsilon, \prec, f_0, f_1, ...)$, some expansion of $H_{\omega_2}$ in a countable language, with $\prec$ giving a well-order. Does there ...
6
votes
0answers
182 views

Large Cardinal Inequalities

Solovay showed that the existence of $0^\dagger$ follows from the existence of two measurable cardinals. We know existence of a measurable cardinals is weaker than existence of $0^\dagger$ so we ...
6
votes
0answers
93 views

On locally small category

Maybe this is a very trivial question for those who are familiar with Set theory. Let $C$ be a category, and $\hat{C}$ be its presheaves. I hear that if both $C$ and $\hat{C}$ are locally small, ...
6
votes
0answers
224 views

The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ $$\begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
6
votes
0answers
230 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
6
votes
0answers
187 views

Largest provably existing ordinal in ZF without power set

If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ...
6
votes
0answers
170 views

Half of a Cohen real

I recently heard from a friend that Zapletal gave a talk at Toronoto where he constructed a proper forcing which adds a real which infinitely often equals every ground model real but doesn't add a ...
6
votes
0answers
105 views

Why can countable recursive constructions not be done using countable choice?

Why can countable recursive constructions not be done using countable choice? For example, replace $Z$ by $\omega$ in the following theorem Why can't one prove it using countable choice? (The proof ...
5
votes
0answers
46 views

Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...
5
votes
0answers
81 views

“How strong is $\Diamond_\kappa^+$?”

In $\text{ZFC}$, we know that $\Diamond_{\kappa}^+ \implies \Diamond_{\kappa}$ and $\Diamond_{\kappa^+} \implies 2^\kappa = \kappa^+$, so that we may think of $\Diamond^+$, i.e. $\Diamond_\kappa^+$ ...
5
votes
0answers
94 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
5
votes
0answers
128 views

Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ...
5
votes
0answers
175 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
5
votes
0answers
334 views

Good Reference for Large Cardinals/Homotopy Theory

I'm planning on attending a conference in Barcelona in September called "Large Cardinal Methods in Homotopy Theory" and want to try to be as prepared as I possibly can. Are there good references for ...
5
votes
0answers
211 views

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of ...
4
votes
0answers
54 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} ...
4
votes
0answers
58 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
4
votes
0answers
69 views

Strange Consequences of Large Cardinals in Probability

Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics. On the other hand we know that if we bring even the least ...
4
votes
0answers
75 views

Do We Need Non-Constructible Sets?

I was reading about Godel's Constructible Universe in which the Continuum Hypothesis and Axiom of Choice are true. It made me wonder, what kind of math would we be unable to do if every set were ...
4
votes
0answers
70 views

Has anyone proposed an axiom for ZFC that implies that many different cardinal characteristics are distinct?

I think it is more interesting when cardinal numbers are distinct, rather than equal. Has anyone proposed an axiom for ZFC that, in one fell swoop, implies that many different cardinal characteristics ...
4
votes
0answers
75 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that ...
4
votes
0answers
56 views

Cardinality of Countable Support Iteration of Proper Forcing

Suppose $\mathsf{CH}$ holds. Suppose $\mathbb{P}$ is a forcing such that $\mathsf{ZFC} \models \mathbb{P} \text{ is proper and } |\mathbb{P}| = 2^{\aleph_0}$. For instance, Sacks forcing is such an ...
4
votes
0answers
56 views

Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition. Definition. Call a set of axioms $\Phi$ in ...
4
votes
0answers
80 views

How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
4
votes
0answers
86 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
4
votes
0answers
62 views

Absoluteness of Satisfaction Relation for Models of the type $J_\alpha$

Is the satisfaction relation absolute between $J_\alpha \subseteq J_\beta$? That is, given a language $L$, a $L$-structures $M$, a formula $\varphi$, and $x$ which are all in $J_\alpha$, is it true ...
4
votes
0answers
52 views

Full Theory of Structure of Set Theory and Generic Extensions

Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of ...
4
votes
0answers
111 views

What axioms are necessary to prove Mostowski Collapse?

I've been reading Schimmerling's A Course on Set Theory, and have been enjoying it a lot so far. However, some times he's less clear than I think he could be. For example, during a proof of Mostowski ...
4
votes
0answers
50 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
4
votes
0answers
86 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
4
votes
0answers
77 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
votes
0answers
94 views

Infinite “String” of Implication Statements

This question is inspired by the conversations at Does this require transfinite induction? First of all, does an infinite string of implication statements have a conclusion? I don't think so, but I ...
4
votes
0answers
81 views

Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
4
votes
0answers
79 views

What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?

Consider the first-order theory obtained by adjoining to ZFC (or, better, Mac Lane set theory) the Grothendieck–Verdier (or Bourbaki?) universe axiom: For each set $x$ there exists a Grothendieck ...
4
votes
0answers
63 views

What is the name of the function that indexes Grothendieck universes?

Assume Tarski-Grothendieck set theory. Then Grothendieck universes form a well-ordered proper class, so we can let $U_\alpha$ denote the $\alpha$'th Grothendieck universe, where $\alpha$ is an ...
4
votes
0answers
2k views

Lists of open problems in set theory

Are there any publicly available lists of open problems in set theory besides the following ones? (And if so, what are they?) http://www.math.wisc.edu/~miller/res/problem.pdf ...