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 ...

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24
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510 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 $M$...
14
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0answers
475 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 ...
13
votes
0answers
315 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 ...
12
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0answers
299 views

Spanning the reals with a small set - choicelessly

Working in ZF (so, no choice): is it possible that there is a set of reals $X$ such that $\vert X\vert<\mathbb{R}$, but $X$ generates $\mathbb{R}$ as a subgroup under addition? This seems ...
10
votes
0answers
99 views

Category-theoretic properties of cardinals

Let $\kappa$ be a cardinal, let $\mathbf{H}_\kappa$ be the set of hereditarily $\kappa$-small sets, and let $\mathbf{Set}_{< \kappa}$ be the full subcategory of $\mathbf{Set}$ corresponding to $\...
10
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0answers
238 views

The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need $\omega_1$-...
10
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0answers
134 views

$\sf ZF$ — Sets that can be proven to exist

There are only countably many formal proofs in $\sf ZF$. Thus, there are only countably many sets that can be proven to exist in $\sf ZF$. This collection of sets seems to satisfy $\sf ZF$'s axioms; ...
10
votes
0answers
134 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 $x$ ...
10
votes
0answers
777 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 ...
10
votes
0answers
391 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 ...
9
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0answers
192 views

An exercise in Fine Structure of constructible universe concerning projectum patterns

This question assumes some familiarity with Jensen's fine structure analysis of the constructible universe L (https://en.wikipedia.org/wiki/Jensen_hierarchy, http://www.math.cmu.edu/~laiken/papers/...
9
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0answers
194 views

A consequence of zero sharp (successor cardinals having countable cofinality)

So the existence of $0^\sharp$ in set theory is really the assertion on the existence of indiscernibles for the constructible universe $L$ that also "generate" $L$ (see http://en.wikipedia.org/wiki/...
9
votes
0answers
409 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 ...
9
votes
0answers
381 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 ...
9
votes
0answers
310 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& \boldsymbol{+}&...
9
votes
0answers
324 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 ...
8
votes
0answers
122 views

What can the reals of an inner model be?

This is probably a silly question. Call a set of reals $X$ a constructibility ideal (in analogy with a Turing ideal) if $X$ is closed under effective join $r\oplus s: n\mapsto 2^{r(n)}3^{s(n)}$ and ...
7
votes
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71 views

Question about a linear order

Let $L$ be the set of countable limit ordinals. For each $\alpha \in L$, let $\langle \alpha_n : n < \omega \rangle$ be a strictly increasing cofinal sequence in $\alpha$. Define a linear order on $...
7
votes
0answers
88 views

Turning ZFC into a free typed algebra

The standard way of using ZFC to encode the rest of mathematics is sometimes criticized because it introduces unnecessary, strange properties such as, for example $1\in 2$ if we encode integers by ...
7
votes
0answers
170 views

Where does the term “mouse” (in set theory) come from?

Does the term mouse stand for "Model Of Universe with Sequences of Extenders" or perhaps mice stands for "Model Including Cardinal Extensions"? A quick review: Gödel's L-universe is a core model ...
7
votes
0answers
146 views

What well-orders are definable over $V$?

Let $V$ be a (the) universe of sets, and $On^V$ denote the ordinals of $V$. It is well known that there are formulae that seem to define orderings `longer than' $On^V$. For example: $\alpha < \...
7
votes
0answers
139 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
7
votes
0answers
98 views

Infinite palindromes in number of nonisomorphic posets is independent of $\mathsf{ZF}$

The following is an "exercise" in P. Stanley's book "Enumerative Combinatorics": Let $f(n)$ be the number of nonisomorphic $n$-element posets (...) let $\mathcal{P}$ denote the statement that ...
7
votes
0answers
122 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 ...
7
votes
0answers
157 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 ...
7
votes
0answers
245 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?
7
votes
0answers
249 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
382 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 ...
6
votes
0answers
72 views

If $U,D$ are $\kappa$-complete nonprincipal ultrafilters on $\kappa$ and $j_U = j_D$, is $U=D$?

Here, $j_U$, $j_D$ are the canonical elementary embeddings induced by the measures $U,D$ on $\kappa$. I actually wish to ask three related questions. Let Ult$_U(V)$, Ult$_D(V)$ be the transitive ...
6
votes
0answers
64 views

Can a Banach measure be consistent with Friedman's Fubini-type theorem for non-measurable functions?

I read on Wikipedia that Harvey Friedman proved that the following is consistent with ZFC+¬CH: For all functions $f:[0,1]^2 \mapsto \mathbb{R}^+$ such that both $\int_0^1 \left ( \int_0^1 f(x,y) dy \...
6
votes
0answers
91 views

Every $f\in\omega^\omega$ is bounded by the “increasing enumeration” of the intersection of a countable dense set and a dense open set in $\mathbb{R}$

I am studying the theorem 2.2.6 of "On the structure of the real line" of book Bartosznky-Judah. In the proof of theorem 2.2.6 the part $(4) \to (5)$ $(4)$ for every family of dense open subsets ...
6
votes
0answers
92 views

Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
6
votes
0answers
170 views

ZF and weakly inaccessible cardinals

This question should probably have been asked 3 years ago (perhaps it has and was removed for some reason?) In 2011, Alexander Kiselev claimed to have proved in ZF that there are no weakly ...
6
votes
0answers
275 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
111 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
249 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
204 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
192 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
83 views

Is 4 really that significant?

I have seen the ZFC theorem, (($2^{\aleph_n}$ $<$ $\aleph_\omega$ for all $n$ $\epsilon$ $\mathbb N$ ) $\rightarrow$ $2^{\aleph_\omega}$ $<$ $\aleph_{\omega_4}$). My question is whether this ...
5
votes
0answers
71 views

Well-ordering of the reals in ZF with constructibility?

The question Do we know that we can't define a well-ordering of the reals? states: There exist pointwise definable models of ZFC where every set is definable without parameters: it is the ...
5
votes
0answers
85 views

Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
5
votes
0answers
91 views

Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
5
votes
0answers
51 views

Notation for inhabited sets

A set $X$ is called inhabited if it has some element. In classical mathematics, this means that it is not the empty set, so that one usually writes $X \neq \emptyset$. However, in intuitionistic ...
5
votes
0answers
148 views

“Clubbiness” of $\Pi^1_n$ sets

I'm sure this is just my google-fu failing me, but: what are sufficient large cardinal axioms to guarantee "Every (boldface) $\Pi^1_n$ set of countable ordinals contains or is disjoint from a club ...
5
votes
0answers
62 views

Vaught's essentially undecidable set theory

Today I was reading this paper, which includes discussion of essential undecidability of various weak theories. On page 24, I was surprised to find out that Vaught has showed that set theory with the ...
5
votes
0answers
93 views

Why does Namba forcing preserve $\omega_1$?

I have seen proofs that under CH, Namba forcing does not add reals, and thus preserves $\omega_1$. How do you prove in ZFC alone that it preserves $\omega_1$? I have also seen the stronger claim ...
5
votes
0answers
71 views

Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
5
votes
0answers
204 views

Cutting a Banach-Tarski Cake

I was reading a cake-cutting problem here (not really related, so I won't link to it), and for some reason, this variation occurred to me. I have no idea whether this problem is even well-formed: ...
5
votes
0answers
60 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
5
votes
0answers
108 views

Many $(\kappa+\alpha)$-strong cardinals below a $(\kappa+\beta)$-strong one for $\alpha < \beta < \kappa$

Let $\lambda$ be an ordinal. A cardinal $\kappa$ is $\lambda$-strong iff there is some inner model $M$ and an elementary embedding $$ j \colon V \rightarrow M $$ s.t. $crit(j) = \kappa$ and $V_\lambda ...