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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3
votes
0answers
63 views

Handwriting cardinal numbers [closed]

In books the cardinal numbers are usually written with a gothic font. Is there any convention how to handwrite cardinal numbers, in particular, how to handwrite the continuum ($\mathfrak c$)
4
votes
1answer
56 views

Do Russell's socks form a Dedekind-finite infinite set?

A countable collection of pairwise disjoint two-element sets $(A_n)_{n<\omega}$ is called Russell-sequence if $\prod_{n<\omega} A_n$ is empty. (That is, there is no way to choose one of two ...
6
votes
1answer
160 views

Countable elementary submodels

I'm having some trouble understanding elementary submodels. Let $H_\chi$ be the set of all sets which are hereditarily of cardinality $<\chi$. Let $\textbf{N}=(N,\in)$ be a countable elementary ...
7
votes
2answers
526 views

Cardinality of the Euclidean topology and the axiom of choice

It is relatively straightforward to prove that the Euclidean topology has the same cardinality as the space itself. I have sketched a proof below. The proof seems to rely quite heavily on the axiom of ...
1
vote
0answers
55 views

Suppes definition of a set

Just beginning to study Suppes "Axiomatic Set Theory" carefully. Early on, in Section 2.2, Definition 1, he gives the following seminal definition of a set: y is a set $\Leftrightarrow$ There exists ...
4
votes
0answers
52 views

Keeping unwanted generic sets out of limit stage of iterated forcing

My question is about keeping unwanted generic sets from appearing at the first limit stage of an iterated forcing. The usual motivation for this is preserving some property $P$ of ZFC models: $P$ ...
13
votes
1answer
250 views

Is there a group with countably many subgroups, but is not countable in ZF?

Inspired by this question, although I don't think it was the OP's intention, hence this separate question: Is there a group $G$ with countably many subgroups, but is a not a countable group itself ...
-1
votes
1answer
163 views

Questions about ZFCU (ZFC with atoms) in second order logic

Is ZFC2U categorical? This is second-order ZFC with full semantics and urelements. Is the cardinality of a set A from ZFC2U expressible by a sentence in second order logic? In 1, because there is ...
1
vote
2answers
66 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
2
votes
0answers
70 views

Solovays model vs “traditional” measure theory

I have just finished my first course in measure theory and I feel I have a good grasp of the concepts of measures, sigma-algebras and the Lebesgue integral. In the next course I am taking there is an ...
2
votes
1answer
91 views

when/who proposed and developed L(R) as a model of ZF?

In trying to understand L($\bf{R}$) better, I've not found an exposition of its origin as a model of ZF. It was around before the end of the 1960s, and one might imagine there was an impetus for ...
2
votes
1answer
66 views

A class that contains itself as an element

Can a class be defined which contains itself as an element? I know its forbidden for a set to contain itself, and I have seen arguments that suggest it's possible for a class to do so but none of ...
-4
votes
2answers
102 views

Maximal model in ZFC (reedited) [closed]

¿does a maximal model exist for ZFC? I wanna mean, that all the other models are subsets of this model. IS it demostrable , undemostrable or undecidable in ZFC? Model in a metamathematical sense
9
votes
1answer
156 views

Countable number of subgroups $\implies $ countable group

I know that if a group $G$ has a finite number of subgroups then the group $G$ is finite. But if a group $G$ has countable number of subgroups then is the group countable?
4
votes
1answer
116 views

Using Compactness to find a non-constructible set

$\newcommand{\ZFC}{\mathit{ZFC}}$I was trying to explain the first ideas of forcing to a friend and I recalled the construction of a model of non-standard arithmetic by using compactness. It is clear ...
2
votes
1answer
72 views

Understanding countable elementary submodels

So I'm having some trouble understanding the existence of countable elementary submodels. I have read and understand the Löwenheim–Skolem theorem, so given a model I understand how to build a ...
2
votes
1answer
73 views

Is the use of the meta-meta-theory allowed in proving an independence result?

I am wondering about the use of a meta-meta-theory in proving statements of the form "$\phi$ is independent of $\mathcal{T}$" in mathematical logic. Short question: Is using a result from the ...
1
vote
1answer
35 views

Comparing Hartogs number to set of all sets of relations on a set

Let $A$ be a set and let $\mathcal{H}(A)$ be the Hartogs number of $A$. Show that $|\mathcal{H}(A)|\neq|\mathcal{P}(\mathcal{P}(A\times A))$. Proof attempt: By contradiction. Suppose that ...
4
votes
1answer
70 views

Proving that the existence of strongly inaccessible cardinals is independent from ZFC?

ZFC can't prove that strongly inaccessible cardinals exist, or else it would prove that a model of itself exists and hence $Con(ZFC)$. So this leaves us with two options: ZFC proves there are no ...
5
votes
0answers
51 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 ...
11
votes
3answers
357 views

Subsets in axiomatic set theory - ZFC

A naive question about axiomatic set theory I'm trying to teach myself some basic set theory by reading Set Theory for the Working Mathematician by Krzysztof Ciesielski, and I'm only in Chapter 1 ...
1
vote
0answers
31 views

About sets and injections [duplicate]

Let $A,B$ be two sets. Is there necessarily an injection from $A$ to $B$ or an injection from $B$ to $A$ ?
5
votes
1answer
54 views

Cardinals and generic extensions

Let us consider a model of ZFC $M$, a forcing $\mathbb{P}\in M$. If $G$ is a $\mathbb{P}$-generic filter over $M$ we can construct the generic extension $M[G]$ of $M$. Given any cardinal number ...
4
votes
1answer
77 views

Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
4
votes
1answer
41 views

The well founded core of a solid $\operatorname{ZF}^-$-model obeys $\Sigma_0$-collection

Let $\mathcal A = (A;E)$ be a model of $\operatorname{ZF}^-$ such that its well-founded core $\operatorname{wfc} (\mathcal A)$ (we also write $\operatorname{wfc}(\mathcal A)$ for its universe) is a ...
18
votes
2answers
337 views

Without the Axiom of Choice, does every infinite field contain a countably infinite subfield?

Earlier today I asked whether every infinite field contains a countably infinite subfield. That question quickly received several positive answers, but the question of whether those answers use the ...
2
votes
3answers
85 views

If a certain kind of object exists and is unique, can we prove its existence and uniqueness without axiom of choice?

Suppose that using Zorn's lemma, we have proven that an object with some properties exists and then we've proven that such object is unique. Can we always conclude that we can prove the ...
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?
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 ...
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 ...
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
69 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
91 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
123 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 ...
-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 ...
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 $\Pi^1_1$ ...
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 ...
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 ...
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 ...
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 ...
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 ...
3
votes
1answer
165 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
2answers
78 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 ...
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 $\langle \mu_\alpha : \alpha \in \lambda\rangle$ ...
0
votes
1answer
37 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?