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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7
votes
1answer
275 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
528 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
838 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
554 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
533 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
827 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
797 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
359 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
498 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
289 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
351 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
258 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
310 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
567 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, ...
29
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
214 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
390 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
517 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
171 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
473 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
761 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
785 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
884 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
625 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
571 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 ...
7
votes
2answers
852 views

How can I write the Axiom of Specification as a sentence?

I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification". To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...
4
votes
2answers
368 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
6
votes
2answers
2k views

Symbol for the cardinality of the continuum

The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not ...
10
votes
2answers
596 views

Surreal and ordinal numbers

Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example: $\omega_0 = < ...
0
votes
1answer
219 views

Initial segments order-isomorphic

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$. ...
4
votes
2answers
291 views

Showing that well-ordered subsets of $P(\omega)$ are countable

I have the following problem: Show that no uncountable subset of $P(\omega)$ is well-ordered by the inclusion relation. I think they want me to do by embedding it in a separable complete dense ...
1
vote
1answer
253 views

Defining $\mathbb{Z}_+$

While reading Munkres' Topology section about integers and reals (Chapter 1, Section 4), he defines the set $\mathbb{Z}_+$ as: Definition: A subset $A$ of the real numbers is said to be ...
11
votes
2answers
2k views

Are there uncountably infinite orders of infinity?

Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...
1
vote
1answer
245 views

Specific equivalent to the Axiom of Choice involving the empty set

I'm trying to remember a particular theorem of ZF but unfortunately my memory is quite incomplete. The theorem is of the form (some set operation) is either (expected answer) or the empty set. If ...
12
votes
2answers
795 views

Transfinite Induction and the Axiom of Choice

My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set? I have read ...
5
votes
1answer
184 views

Where in the analytic hierarchy does V=L start having consequences?

I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable ...
6
votes
3answers
1k views

Unary intersection of the empty set

In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, ...