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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1answer
263 views

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I'm having trouble understanding the statement: If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well. I understand the ...
2
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4answers
219 views

What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions?

I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\mathbb{N}$ be two monotonicly ...
25
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2answers
1k views

Is the fundamental theorem of calculus independent of ZF?

By the fundamental theorem of calculus I mean the following. Theorem: Let $B$ be a Banach space and $f : [a, b] \to B$ be a continuously differentiable function (this means that we can write $f(x + h)...
4
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3answers
225 views

What is the essential difference between the terminologies “associative” and “semi-group”.

Can only a map $$*:S\times S\rightarrow S$$ be associative? If I look at $$(a* b)* c=a* (b* c),$$ then it seems I have to rule out the more general case $$*:A\times B\rightarrow C.$$ But $...
5
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2answers
387 views

How does the supremum of the cardinalities of a set of ordinals relate to the supremum of the ordinals?

Suppose we have a set of ordinals such that their supremum is a cardinal (not in the original set). $\sup\{\alpha\}=\kappa$ I'm interested in the supremum of the cardinalities of those ordinals: $\...
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3answers
717 views

Is the set of all definable subsets of the natural numbers recursively enumerable?

I asked myself similar questions before, for example "Are the definable real numbers countable"? It seemed to me that the set of all explicitly and unambiguously definable objects is "countable", ...
11
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3answers
654 views

Are surreal numbers actually well-defined in ZFC?

Thinking about surreal numbers, I've now got doubts that they are actually well-defined in ZFC. Here's my reasoning: The first thing to notice is that the surreal numbers (assuming they are well ...
7
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1answer
474 views

What is $\aleph_0$ powered to $\aleph_0$?

By definition $\aleph_1 = 2 ^{\aleph_0}$. And since $2 < \aleph_0$, then $2^{\aleph_0} = {\aleph_1} \le \aleph_0 ^ {\aleph_0}$. However, I do not know what exactly $\aleph_0 ^ {\aleph_0}$ is or how ...
4
votes
1answer
260 views

Are there ways to describe the Martin Axiom intuitively?

I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking ...
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2answers
664 views

(ZF)subsequence convergent to a limit point of a sequence

Arthur's answer; (ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed. Let $\{p_n\}$ be a sequence in a metric space $X$. Let $B=\{p_n|n\in\mathbb{N}\}$ and $...
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2answers
1k views

What is actually “relatively consistent”?

Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc. However, why does this ...
4
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1answer
118 views

Proving $|A^A|=|2^A|$ for infinite $A$. [duplicate]

Possible Duplicate: Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$? How can one prove that $|A^A|=|2^A|$ for infinite $A$? (summary of proof or providing link with proof will ...
1
vote
1answer
348 views

Totally ordered set with greater cardinality than the continuum

Does there exist a totally ordered set $S$ with cardinality greater than that of the real numbers? Sequences are continuous functions with domain $\mathbb{N}$ and paths are continuous functions with ...
21
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9answers
1k views

What is it about modern set theory that prevents us from defining the set of all sets which are not members of themselves?

We can clearly define a set of sets. I feel intuitively like we ought to define sets which do contain themselves; the set of all sets which contain sets as elements, for instance. Does that set ...
2
votes
4answers
305 views

An infinite set having “one more element” than another infinite set

A classic example of homeomorphism is between a sphere missing one point and a plane To see this, place a sphere on the plane so that the sphere is tangent to the plane. Given any point in the plane, ...
6
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2answers
246 views

Why bother proving that a class is a set?

Fix a set theory, say ZFC. Now suppose you have a statement $P(x)$ written in the language of that theory, which has precisely one free variable $x$. We can conceive of the class of all $x$ in the ...
3
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1answer
204 views

When collapsing a cardinal, what ordinal does it become?

Work in $V$. Let $P = \text{Col}(\omega, \omega_1)$ and suppose that $G$ is generic for $P$ over $V$. Then $V[G]\models |\omega_1^V|=\aleph_0$ and $\omega_2^V=\aleph_1$. In particular, $V[G]\models\...
4
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4answers
435 views

(ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior

Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF. Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$. ...
3
votes
3answers
225 views

Class of manifolds is a set?

Is the class of all 2-countable manifolds a set? I think so: each such space is a countable union of sets of cardinality $|\mathbb{R}^n|\!=\!|\mathbb{R}|$, i.e. a manifold has cardinality continuum, ...
9
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2answers
291 views

$\varepsilon$-number countability without choice

Let $\alpha\mapsto\varepsilon_\alpha$ be the enumeration of the $\varepsilon$-numbers--that is, those $\alpha$ such that $\omega^\alpha=\alpha$--by the ordinals. If we know that countable unions of ...
7
votes
1answer
115 views

Existence of a bijection without AC

Let $A$ and $B$ be subsets of a set $X$ and let $f:A\rightarrow X\setminus A$ and $g:B\rightarrow X\setminus B$ be bijections. Is it possible to show without AC that there is a bijection $h:A\...
3
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3answers
376 views

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$) I'm struggling with this problem for a week. I'm talking about this in Metric space. Here's the part of argument ...
6
votes
2answers
167 views

Sum of cardinals without AC

Let $A$ and $B$ be infinite sets. To show $|A\cup B|=\max\{|A|,|B|\}$ we need AC. Now let us assume $|A|<|B|$. Can we show $|A\cup B|=|B|$ without AC?
0
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1answer
355 views

Is cardinality a total order? Is AC necessary? [duplicate]

Possible Duplicate: Is the class of cardinals totally ordered? Intuitively, it seems like for any sets $A,B$ either $\lvert A\rvert\leq \lvert B\rvert$ or $\lvert B \rvert \leq \lvert A\rvert$....
10
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1answer
325 views

Shelah Cardinals and Modern Set Theory

Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?
2
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1answer
1k views

Cardinality of cartesian square

Given an infinite set $A$ - does the cardinality of $A$ equal to the cardinality of $A^2$?
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1answer
108 views

(ZF) Dedekind infinite + Limit Point Compact ⇒ Separable

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF) Yesterday, i posted this question and got an answer that 'Limit Point Compact⇒Separable' is unprovable ...
4
votes
1answer
123 views

Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$

This is an exercise from Kunen's book. Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$ are equal. What I've tried: I want to prove by using induction on $m$. ...
5
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2answers
416 views

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF)

I can prove this in ZFC, but don't know how to prove this in ZF. Following is the argument of this in ZFC. Fix $0<r\in \mathbb{R}$ and $x_0\in X$. Let $A_i = \{x\in X\mid d(x,x_j)\ge r,\, j<i\}$...
2
votes
1answer
281 views

Play a slot machine for uncountable number of times

There is a slot machine. You insert one coin, it destroy the coin and return countable number of coins. Then you can pick any one of the returned coin and put in the slot machine again. Formally, ...
4
votes
2answers
231 views

Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number

Firstly, I give the definition of the epsilon number: $\alpha$ is called an epsilon number iff $\omega^\alpha=\alpha$. Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an ...
5
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2answers
1k views

some elementary questions about cardinality

I know little about set theory and while reading some Algebra proof I had difficulty on some details. So my questions are : If $X$ is an infinite set and $Y$ is the set of all finite subsets of $X$, ...
5
votes
2answers
333 views

What interpretations exists for the singleton of the singleton of the empty set $\{\{\varnothing\}\}$ and its singleton $\{\{\{\varnothing\}\}\}$?

What I know I know from Peano's axioms that the empty set is equivalent to the natural number $0$ and that the singleton of the empty set is equivalent to the natural number $1$. ( http://www....
5
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2answers
323 views

Equivalent characterizations of ordinals of the form $\omega^\delta$

Let $\alpha$ be a limit ordinal. Show the following are equivalent: $\forall \beta, \gamma<\alpha (\beta+\gamma<\alpha)$ $\forall \beta<\alpha(\beta+\alpha=\alpha)$ $\forall X\subset \alpha(...
2
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1answer
254 views

How do I prove the existence of infinite union in ZFC?

Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?
1
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1answer
127 views

How to show the two inequalities?

This is an exercise from Kunen's book. Show that $\alpha < \beta$ implies that $\gamma+\alpha<\gamma+\beta$ and $\alpha+\gamma\le\beta+\gamma$. Given an example to show that the "$\le$" cannot ...
4
votes
1answer
519 views

Is a countable product of compact intervals in $\mathbf R$ compact (without using the AC)?

Let $\{I_n=[a_n,b_n]\}_{n\in\mathbf N}$ be a countable collection of closed, bounded intervals in $\mathbf R$. Is the infinite Cartesian product $$\prod_{n=1}^\infty I_n$$ compact without using the ...
6
votes
1answer
1k views

Proving that the set of algebraic numbers is countable without AC

A complex number $z$ is said to be algebraic if there is a finite collection of integers $\{a_i\}_{i\in n+1}$, not all zero, such that $a_0z^n + … + a_n = 0$. Then can I prove the set of all ...
3
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1answer
722 views

What does the continuum hypothesis imply?

Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom? I'm sorry if this question was already asked. I'm also sorry if there ...
3
votes
3answers
117 views

Does the Cartesian product get smaller if I use fewer sets?

My introduction into Axiom of Choice has been kind of confusing (Zorn's lemma) for the start, so it took me some time to realize it's nothing but to say The non-empty product of non-empty sets is non-...
1
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1answer
87 views

Name for a set which has an order?

As we all know, a set is a collection of elements which have no particular order and no multiplicity. So what do you call a construct which does store its elements in a specific order? What is the ...
5
votes
1answer
492 views

Definition of Cantor Set without AC

You can see the original text that i thought AC is used here; From Walter Rudin: Principles of Mathematical Analysis, 3rd ed., ISBN 0-07-054235-X, p.41-42. 2.44 The Cantor set The set which we ...
23
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1answer
933 views

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product ...
2
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2answers
62 views

Cocountable fibers

Let $C$ be an uncountable set. Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
3
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2answers
328 views

True Statement in ZF is true statement in ZFC?

How do we know an $ \aleph_1 $ exists at all? Here, you can see that $\aleph_2≦2^{\aleph_0}$, which is a contradiction to CH in ZFC. So if 'true statements in ZF is true in ZFC' is true, CH must be ...
5
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1answer
253 views

Stronger than ZF, weaker than ZFC

Can you please name axiom system that is strictly weaker than ZFC and strictly stronger than ZF? (such as DC, AC$_\omega$) I searched for it but i could only find these two. If there are statements ...
3
votes
1answer
141 views

Which infinite cardinals can be defined using partition relations?

Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then $\kappa$ is weakly compact if $\kappa$ satisfies $\kappa\to(\kappa)...
2
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1answer
124 views

An inequality in a proof of Kunen's Inconsistency

This may be a silly question, but I keep coming back to it. Let $j:V\prec M$ be a non-trivial elementary embedding with $M$ a transitive class and $\kappa$ the critical point of $j$. Define the ...
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0answers
98 views

Infinitesimals and infinite elements among the transseries

In the quest for extensions of $\mathbb{R}$ and $\mathbb{C}$ that contains infinitesimals, infinities (and even more exotic beasts like $\omega - 1$ and $\sqrt{\omega}$) I came across the theory of ...
2
votes
1answer
142 views

A characteristic of intersection with cartesian product

Fix some binary relation $f$. Does there necessarily exist a set $C$ such that $(x\times x)\cap f\ne \varnothing \Leftrightarrow x\cap C\ne \varnothing$ for all sets $x$?