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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4
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2answers
354 views

Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set $x$ contains a member $y$ ...
0
votes
1answer
51 views

Consistency of the Subcomplete Forcing Axiom (relative to a supercompact cardinal)

In the introduction to his Singapore lecture Jensen mentions that the Subcomplete Forcing Axiom is consistent relative to a supercompact cardinal. Can anyone refer me to a proof of this claim?
8
votes
1answer
144 views

Transitive models and CH

Suppose $M, N$ are two countable transitive models of ZFC which have same ordinals, cofinalities and reals (but not necessarily same sets of reals!). Suppose $M$ models the continuum hypothesis. Can ...
3
votes
3answers
66 views

Does the following define a Mahlo cardinal?

Let M be a cardinal with the following properties: - M is regular - $\kappa < M \implies 2^\kappa < M$ - $\kappa < M \implies s(\kappa) < M$ where $s(\kappa)$ is the smallest strongly ...
0
votes
0answers
43 views

Obstructions to non-separable geometry

I'm curious about generalizations of Euclidean geometry to non-separable spaces. Specifically, my question would be: Which are the obvious obstructions to having a model of (some considerable ...
1
vote
1answer
46 views

Fields of intermediate cardinality (2)

Assuming the existence of a cardinal $\aleph_0 <\mathfrak{m} < 2^{\aleph_0}$, does it follow that there is a subfield of $\mathbb{R}$ of cardinality $\mathfrak{m}$ containing no algebraic ...
1
vote
2answers
83 views

Fields of intermediate cardinality

Assuming the existence of a cardinal $\aleph_0 <\mathfrak{m} < 2^{\aleph_0}$, does it follow that there is a subfield of $\mathbb{R}$ of cardinality $\mathfrak{m}$?
1
vote
2answers
31 views

Uncountable set of functions without a countable set of generators

Is there a subset $\mathfrak{F}$ of $\omega^\omega$, of cardinality $\aleph_1$, such that no subset $\mathfrak{G}$ of $\mathfrak{F}$ of cardinality $<\aleph_1$ generates $\mathfrak{F}$ in the sense ...
3
votes
1answer
68 views

Elementarily equivalent forcing extension?

Is it possible to take a forcing extension which is elementarily equivalent to the ground model? Here I'm assuming the extension is proper, that is, it adds a new set. It's clear it can't be an ...
2
votes
2answers
90 views

On Fraenkel-Mostowski choiceless set theory

I have been trying to solve an exercise from Kunen (1980) on Fraenkel and Mostowski's construction of a choiceless model of set theory. I have a couple of questions: The model is constructed from ...
1
vote
0answers
47 views

Absoluteness of Relations for Transitive Models

I am wondering if it's true in general that $ ( A \text{ is an antichain in } P)^{ M} \Leftrightarrow ( A \text{ is an antichain in } P)$ whenever $M$ is a transitive ZFC model. I recall that ``being ...
3
votes
2answers
122 views

Minimal foundations for Cardinal Arithmetic

I would like to develop a theory of cardinal numbers that relies on as weak a basis as possible. Therefore, I would like to know if there is a way to even define a cardinal number for every set ...
9
votes
1answer
229 views

Product of metric outer measures

The problem below has been asked recently already but, as a naive user, I got burned (well singed perhaps) because I asked the question in the wrong place. So if this looks like a redundant question ...
-1
votes
1answer
107 views

Reflection Principle vs. Löwenheim-Skolem-Theorem

From my undestanding a standard method of deducing relative consistency results is the following: By a combination of the Levy reflection principle, Skolem-Hulls and Mostowski Collapse we show: If ...
2
votes
2answers
91 views

Non analytic ideals on $\omega$

I would like to gather examples of NON analytic ideals on $\omega$. However, I have found nothing in the books and papers I have consulted. Could anyone tell me some reference/s?
0
votes
3answers
140 views

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like ...
3
votes
1answer
75 views

$\Pi^1_1$ singletons and $\Delta^1_2$ wellorders on $\omega$ in $L$

I have been trying to show the supremum $\delta^1_2$ of ordinals that are $\Delta^1_2$ wellorders on $\omega$ is exactly equal to the least ordinal $\delta$ such that $L_\delta$ contain all $\Pi^1_1$ ...
1
vote
1answer
59 views

Show that every proper filter on a set X can be extended to a proper prime filter?

Are the following enough to complete the proof? The union of a chain of filters is a filter. A maximal filter is an ultra-filter. How I can use Zorn's lemma to find the maximal filter?
1
vote
2answers
77 views

Where to go after Halmos' *Naive Set Theory*

I'm in the process of finishing Halmos' Naive Set Theory, and I found the subject fascinating, so I would like to carry on reading about Set Theory when I'm done. From what I've been able to gather ...
1
vote
0answers
34 views

The Axiom of Choice, and the statement that whenever there is a surjection $A \to B$ there is an injection $B \to A$ [duplicate]

Is there a model of $\mathsf{ZF}$ in which the Axiom of Choice fails but the following statement holds for arbitrary sets $A, B$: there is an injection $A \to B$ if there is a surjection $B \to ...
7
votes
1answer
309 views

Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?

Set theory is typically formalized as a one-sorted theory without urelements. Is it possible to do the same with category theory or higher category theory, formalizing the whole affair as a theory ...
4
votes
1answer
130 views

Is there a countable transitive model satisfying the same set of first-order sentences as $V$? [duplicate]

This is probably a pretty simple question, but I'm tying myself in knots over it. We're all familiar with the Reflection Theorem, Lowenheim-Skolem Theorem, and Mostowski Collapse Lemma for getting ...
3
votes
1answer
164 views

$\mathbb{R}^3$ \ $\mathbb{Q}^3$ is union of disjoint lines. The lines are not in an axis diretion.

I have the following question: $\mathbb R^3\setminus \mathbb Q^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski. ...
1
vote
0answers
42 views

Forcing, $ p \Vdash q \in \dot{G} \Rightarrow p \leq q $

I was wondering if a poset is separative if $ p \Vdash q \in \dot{G} ~~ \Rightarrow p \leq q$ I think it's clear that $ p, q \in G $ and hence are compatible but I am not seeing why ( if it's true ...
4
votes
2answers
100 views

On singular products of cardinal numbers

I want to know whether it is possible to show in $ZFC$ that there exist a limit ordinal $\lambda$, a strictly increasing sequence of cardinal numbers $\langle \mu_\alpha : \alpha \in \lambda\rangle$ ...
1
vote
0answers
66 views

$\mathbb{R}^3$ \ $\mathbb{Q}^3$ is a union of disjoint lines, The lines are not in an axis direction. [duplicate]

I have a folowing question: $\mathbb{R}^3$ \ $\mathbb{Q}^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski. Any can give ...
-1
votes
1answer
69 views

Examples of non-trivial determiner formulas of trnsitive models of ZFC

Notation: For each $\{\in\}$-formula $\varphi(x_1,\cdots,x_n)$ and each $\in$-model $M$, define: $$\varphi (M)=\{(a_{1},\cdots,a_{n})\in M^{n}~|~\langle M,\in\rangle\models \varphi ...
20
votes
1answer
696 views

Are there areas of mathematics (current or future) that cannot be formalized in set theory?

I often read that ZFC can formalize "most" of everyday mathematics, but I could never find an example which it cannot. The closest I got is differential geometry (DF), where some article mentions that ...
0
votes
1answer
36 views

Axiom of Choice with randomized choice function

Does the axiom of choice require the choice function to be deterministic or can it be a random function (i.e., its image under some probability space is the set under considertation?)
2
votes
1answer
73 views

How many objects are in $\mathbf{Set}$? [closed]

... or does this question even make sense, considering the object "collection" of $\mathbf{Set}$ is a proper class rather than a set?
3
votes
2answers
203 views

Why is this contradiction using axiom of constructibility incorrect?

Today I thought of this paradox and I'm trying to find the wrong assumption that causes it. Does anyone know what is wrong in the following argument: let $$A=\{x\in \mathbb{R}|\exists\phi:\forall ...
0
votes
1answer
60 views

$ V_{ \kappa}$ ( $ \kappa $ inaccessible ) models there is a countable model of ZFC

I think that this statement is very well-known but I am a bit unclear on some of the reasoning. I am aware that $ V_{ \kappa }$ models ZFC when $ \kappa $ is an inaccessible cardinal. Therefore by ...
1
vote
0answers
41 views

Does subsets of a semigroup with strictly smaller cardinality insure infinitely many disjoint translation copies of the set?

Let $S$ be a semigroup with infinite cardinality, $A\subset S$ with $|A|<|S|$. Under what condition we may find a infinite net $\{s_\alpha; \alpha\in \Gamma\}$, such that $s_\alpha A \cap s_\beta A ...
1
vote
1answer
30 views

Is $|2^\omega \cap L| = |(\omega_1)^L|$?

Is $|2^\omega \cap L| = |(\omega_1)^L|$? Here I mean the set of all subsets of $\omega$ by $2^\omega$, but I hope that choosing that particular interpretation does not really matter here, as always. ...
2
votes
1answer
111 views

Is there is limit to what $2^{\aleph_0}$ can be in a ctm?

Is there is limit to what $2^{\aleph_0}$ can be in a countable transitive model? How large can be the value of the continuum in a countable transitive (standard) model of ZFC? For instance, if we ...
2
votes
1answer
80 views

If $\kappa$ is regular, is $\prod^{\text{fin}}_{\alpha<\kappa}\operatorname{Fn}(\omega,\alpha)$ $\kappa$-cc?

Given an ordinal $\alpha$ let $\operatorname{Fn}(\omega, \alpha)$ be the set finite partial functions from $\omega$ to $\alpha$. Given a cardinal $\kappa$ let ...
0
votes
2answers
61 views

on the continuum hypothesis: only finitely many cardinals between $\aleph_0$ and $2^{\aleph_0}$

Is it at least known that there are only finitely many cardinals between $\aleph_0$ and $2^{\aleph_0}$?
0
votes
1answer
42 views

Showing extensionality for Mostowski collapse

I have been trying to show that if $\gamma$ is such that there is a real (an element of $2^\omega$) in $L_{\gamma+1} \setminus L_\gamma$, then there are countable $M$ and a surjection $f : \omega ...
4
votes
1answer
185 views

On the Axiom of Choice for Conglomerates and Skeletons

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and ...
2
votes
1answer
79 views

($\lozenge$) There exists Suslin trees such that their product is Suslin too.

I'm trying to show that under $\lozenge$ assumption there exists $S$ and $T$ Suslin trees such that $S\otimes T$ is also Suslin. I really have no idea how to use the existence of a ...
1
vote
2answers
118 views

Real numbers for beginners

I am thinking about the Wikipedia (I understand disputed) article about “definable real numbers”. It begins to say that, A real number $a$ is first-order definable in the language of set theory, ...
2
votes
1answer
36 views

Measure theory Vitali nonmeasurable set.

On $[0,1]$ I've got the relation $\sim$ defined as: $x \sim y \iff x-y \in \mathbb Q$ this is a relation of equivalence and so: we can make a factor class ...
1
vote
2answers
61 views

König's theorem (set theory) implication

How does König's theorem imply $\quad\aleph_{\omega} \neq \beth_1$?
9
votes
4answers
146 views

Does it make sense to define $ \aleph_{\infty}=\lim\limits_{n\to\infty}\aleph_n $? Is its cardinality “infinitely infinite”?

I recently read a book about infinity, which introduced the basic notions of different kinds of infinity. I'm a total layman concerning this topic, and one question fascinated me: Can we, in some ...
5
votes
1answer
195 views

More than the real numbers:hyperreals, superreals, surreals …?

I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ...
3
votes
2answers
61 views

Countable Elementary Submodels $M \preccurlyeq H ( \theta) $.

I am reading through some of Kunen's material on Elementary Submodels and am a little unclear on one proof. Here is a part of the claim: Let $ \theta$ be an uncountable cardinal, and let $ M$ be ...
4
votes
0answers
68 views

Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
9
votes
1answer
81 views

If a set $S$ has a choice function, does $\bigcup S$ have one too?

I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have ...
2
votes
1answer
70 views

Is there a sequence of universes in TG?

Assume we are working in ZFC + Tarski's axiom (Every set is an element of some universe). I wonder, if there is a universe $U$ with a sequence $(U_n)_{n\in \mathbb{N}}$ in $U$, s.t. $U_1,U_2,\dots$ ...
1
vote
2answers
88 views

Proving infinity vs Axiom of infinity

I am not much of a set theorist, I deal primarely with algebra in my interest and what I study so this is toward set theorists. I am curious as to why cannot infinity be properly proven to exist? I ...