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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8
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2answers
392 views

What does Martin's Maximum imply for $P(\mathbb{R})$?

Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in ...
4
votes
1answer
190 views

Closed subset of stationary set and AC questions

Hope I'm not spamming too much by asking questions on separate threads. I have 2 more questions, not connected one to another, in any way: 1. Show that every stationary set in $\aleph_1$, contains, ...
9
votes
1answer
306 views

Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?

The question contains 2 stages: Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$ This one is pretty clear by induction and by applying Hausdorff's formula. Prove ...
5
votes
0answers
229 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 ...
5
votes
2answers
1k views

Some questions in Set Theory

I have some exam questions that were left unanswered for me: Suppose that for every $\alpha<\kappa$ there is a subset $A_\alpha$ of $\kappa$ of cardinality $\kappa$. Show that there is a subset X ...
12
votes
1answer
775 views

bound on the cardinality of the continuum? I hope not

Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for ...
1
vote
4answers
299 views

Does the reflexivity mean maximum element in partial ordering $x \leq x$ for every $x \in S$

The notation of my teacher confuses me in the title. If x is the same on the both sides, it seems trivial. So I suspect the x is ...
19
votes
1answer
865 views

Bijection between $2^{\mathbb R}$ and $\mathbb{ R ^ R}$

I'm well aware of the standard proof based on cardinality arithmetic to show that these two sets have the same cardinality and the question of defining a bijection between the two sets came up. I ...
8
votes
1answer
907 views

Polish Spaces and the Hilbert Cube

I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : ...
13
votes
4answers
1k views

Is there such a thing as a countable set with an uncountable subset?

Is there such a thing as a countable set with an uncountable subset? Actually I know the answer. Well, I believe I know the answer, which is NO. Unfortunately, the professor in a Theory of ...
5
votes
2answers
382 views

Notation for surreal numbers

On the sound of sounding ridiculous, but in the line of "There are no stupid quetsions": Is there a way to express $\omega_1$ (and in general $\omega_k$ with $k >= 1$ as a Conway game (that is ...
3
votes
1answer
239 views

Rudin–Keisler equivalence

Wikipedia says that if $a\le b$ and $b\le a$ in Rudin–Keisler order for ultrafilters $a$ and $b$, then $a$ and $b$ are Rudin–Keisler equivalent. How to prove this?
7
votes
1answer
277 views

Forcing cardinality of a set

I'm studying Shelah's proof (actually written by Uri Abraham) that adding one generic real implies the existence of a Suslin tree (available in this link, I think that freely for everyone.) The ...
10
votes
1answer
1k views

Complex logic puzzle

This is a puzzle that was sent to me a while back, I am told it is really hard, but supposedly solvable, I cant solve it, but I am interested in the solution, or any tips on how to proceed. In ...
11
votes
2answers
1k views

Uncountable subset with uncountable complement, without the Axiom of Choice

Let $X$ be a set and consider the collection $\mathcal{A}(X)$ of countable or cocountable subsets of $X$, that is, $E \in \mathcal{A}(X)$ if $E$ is countable or $X-E$ is countable. If $X$ is ...
1
vote
3answers
546 views

Set Theory as it Relates to Number Systems?

I've been referred to this website, hopefully you have the background in set theory to help me out here. Got two questions, the first is on number systems arising out of set theory and the second ...
13
votes
4answers
847 views

Is there a statement whose undecidability is undecidable?

We know there are statements that are undecidable/independent of ZFC. Can there be a statement S, such that (ZFC $\not\vdash$ S and ZFC $\not\vdash$ ~S) is undecidable?
9
votes
3answers
559 views

Axiom of Choice and the cardinality of the reals

Assuming the Axiom of Choice, (it seems that) there is a bijection between $\mathbb{R}$ and $\mathbb{N}$ that follows from any well-ordering of the reals. That is, given a well-ordering of ...
4
votes
4answers
536 views

3rd axiom of probability for discrete distribution

it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed. I argue that in the case of a discrete distribution, say a ...
14
votes
1answer
2k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...
7
votes
2answers
861 views

Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice

Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...
11
votes
2answers
811 views

Equivalent statements of the Axiom of Choice

As a little project for myself this winter break, I'm trying to go through as much of Enderton's Elements of Set Theory as I can. I hit a snag trying to show two forms of the Axiom of Choice are ...
5
votes
1answer
360 views

Normal ultrafilters and Stationary sets

If $\kappa$ is a measurable cardinal, and $\mathcal{U}$ is a normal ultrafilter which is $\kappa$-complete then $\mathcal{U}$ extends the club filter (i.e. every club is a member of $\mathcal{U}$). ...
3
votes
1answer
508 views

Dimension of the sequence space and its dual, depending on status of (AC) and (CH)

Let's consider the sequence space $E =\mathbb R^{\mathbb N}$. If I believe in Choice, I have an isomorphism $E \simeq \mathbb R^{(\mathfrak c)}$ for some cardinal $\mathfrak c$. I further have some ...
8
votes
4answers
1k views

Simple example of uncountable ordinal

Can you make a simple example of an uncountable ordinal? With simple I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an ...
3
votes
2answers
365 views

Problem in set theory

the problem is the following: For all $x \in \mathbb{R}$, we assign a finite set $\Phi(x) \subset \mathbb{R} - \{ x \}$. We say that a set $S \subset \mathbb{R}$ is independent if for any $x,y \in ...
4
votes
3answers
292 views

Defining “Small Classes.”

Question: What is the definition of a small class? Is there no such thing as a set which contains other sets such as {{1,2},{1},{2}, 1, 2} (ie, is this really called a "class")? What are some ...
2
votes
1answer
359 views

Normal ultrafilters on measurable cardinals

Let $\kappa$ be a measurable cardinal, we say that $\mathcal{D}$ is a normal ultrafilter iff whenever $g\in\kappa^\kappa$ such that $g<_\mathcal{D} Id$, we have some $\alpha<\kappa$ such that ...
2
votes
2answers
262 views

Splitting a club set

Suppose $\kappa$ is an uncountable cardinal. Then $S\subseteq\kappa$ is club (CLosed UnBounded) if $S$ is unbounded in $\kappa$ and is a closed subset of $\kappa$ under the order topology. My question ...
4
votes
3answers
2k views

Proof that a set is infinite if and only if it has an infinite proper subset

I read somewhere that a set is infinite if and only if it has a proper infinite subset. I also remember seeing someones name attached to this theorem on Wikipedia once, but I can't even find that ...
3
votes
2answers
318 views

“Homomorphism” from set of sequences to cardinals?

First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit ...
3
votes
1answer
573 views

Set theoretic definition of a Natural Number

I am unable to understand the motivation behind the set theoretic definition of a natural number. The definition given in the book by Goldrei is as follows: First he defines an inductive set: A set ...
0
votes
2answers
1k views

Mathematical representation of the largest element in a set

I've looked but cannot find the mathematical way to represent the following: r = Max(x1, x2, x3) I want to mathematically show that r = max value of the set (x1, ...
30
votes
3answers
1k views

Set Theoretic Definition of Numbers

I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural ...
2
votes
1answer
221 views

Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary?

Is the intersection of a decreasing sequence of countably many stationary subsets of $\omega_1$ always stationary? It seems elementary, but I fail to find the answer in textbooks.
4
votes
1answer
394 views

How to prove $\{t\} \notin t$

How to prove $\{t\} \notin t$ using the axiom of foundation (aka axiom of regularity): $A = \emptyset \vee \exists x \in A \forall y \in x : y \not\in A$
9
votes
2answers
531 views

Proving $V_{\kappa}$ is a model of ZFC for inaccessible $\kappa$

Prove that if $\kappa$ is an inaccessible cardinal, then $V_{\kappa}$ satisfies all the axioms of ZFC. How is this done for the axiom of choice and for regularity?
3
votes
1answer
123 views

Ordering on pairs

Define $(a,b) < (a',b')$ if $\max(a,b) < \max(a',b')$ or $\max(a,b) = \max(a',b')$ and $b < b'$ or $\max(a,b) = \max(a',b')$ and $b = b'$ and $a < ...
4
votes
1answer
190 views

$\aleph_0$ to $\aleph_0$

Let $\omega$ denote the countably infinite cardinality and $A$ a cardinality that is strictly larger (i.e., an uncountable one). Is it true that $A^\omega$ has strictly larger cardinality than $A$?
3
votes
2answers
173 views

Recursive definition isomorphism

If $(X, \lt)$ is a well-ordering I can show by transfinite recursion over the ordinals that the function $f(x) = \text{ran} f |_{\hat{x}}$ exists (where $\hat{x} = \{ y : y \lt x\}$). I have obtained ...
4
votes
1answer
168 views

Recursion on ordinals

I have some kind of "homework" question. I have the following theorem: Theorem (Transfinite Recursion over the class of of ordinals $\mathbf{ON}$: Let $\mathbf{V}$ be the class of all sets. If ...
6
votes
3answers
478 views

Multiplying Cardinal Numbers

I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ ...
7
votes
9answers
783 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
12
votes
2answers
801 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
6
votes
1answer
905 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
2
votes
3answers
627 views

Are there many more rational numbers than integers?

The title is somewhat deceptive: I know that $|\mathbb{Q}|=|\mathbb{Z}|.$ But suppose I wanted to compare the sets, knowing that they are of the same cardinality but still wondering if there was ...
2
votes
1answer
228 views

Limiting set theory using symmetry

If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For ...
62
votes
6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
19
votes
5answers
2k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
6
votes
1answer
597 views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...