# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 $\lvert B \rvert \leq \lvert A\rvert$....
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### 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?
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### 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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### (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 ...
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### 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 $m$. ...
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### 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 $A_i = \{x\in X\mid d(x,x_j)\ge r,\, j<i\}$...
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### 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, ...
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### 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 ...
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### 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 $X$, ...
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### 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....
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### 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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### 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 ...
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### 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 $x$?