# Tagged Questions

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

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### What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
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### Is the sum of all natural numbers countable?

I do not even know if the question makes sense. The point is rather simply. If I have the sum of all natural numbers, $$\sum_{n\in \mathbb{N}}n$$ this is clearly "equal to infinity". But since ...
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### What's “the catch” in this question?

I am solving old exam questions and I came across this question: Let $\langle A_n \mid n < \omega\rangle$ disjoint sets such that $\bigcup_{n < \omega}A_n = \mathbb{R}$. Prove that there ...
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### Is the class of cardinals totally ordered?

In a Wikipedia article http://en.wikipedia.org/wiki/Aleph_number#Aleph-one I encountered the following sentence: "If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...
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### Is the axiom of choice needed to show that $a^2=a$?

A comment on this answer states that choice is needed for the statement that $a^2=a$ for all infinite cardinals $a$. In Thomas Jech's Set Theory (3rd edition), his theorem 3.5 proves this statement ...
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### Is there an infinite countable $\sigma$-algebra on an uncountable set

Let $\Omega$ be a set. If $\Omega$ is finite, then any $\sigma$-algebra on $\Omega$ is finite. If $\Omega$ is infinite and countable, a $\sigma$-algebra on $\Omega$ cannot be infinite and countable....
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### Why does $\omega$ have the same cardinality in every (transitive) model of ZF?

As in the title: Why does $\omega$ have the same cardinality in every (transitive) model of ZF? I've been thinking about this for some time now. Can someone show me how to show this by showing me a ...
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### Tightness and countable intersection of neighborhoods

The following is a problem a colleague has encountered. He would like to know whether the following conjecture is right, wrong, or neither: Let $X$ be a topological space of countable tightness ...
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### Existence of a sequence that has every element of $\mathbb N$ infinite number of times

I was wondering if a sequence that has every element of $\mathbb N$ infinite number of times exists ($\mathbb N$ includes $0$). It feels like it should, but I just have a few doubts. Like, assume ...
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### The largest number system

If my number set construction memory doesn't fail me (I'll edit if errors are pointed out), we start out with Peano's axioms to get to $\mathbb{N}$, and in the need of an additive inverse for its ...
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### Infinite sets and their Cardinality

(I am a 13 year old so when you answer please don't use things that are TOO hard even though I actually can understand quite complex stuff) I was studying Infinite sets and their cardinality (not in ...
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### Cardinality of the infinite sets

Consider the following problem: Which of the following sets has the greatest cardinality? A. ${\mathbb R}$ B. The set of all functions from ${\mathbb Z}$ to ${\mathbb Z}$ C. The ...
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### axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
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### Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?
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### Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$?

Is it possible that $2^n=3^n$ for some Dedekind-finite cardinal $n\gt0$? I think the question speaks for itself, but let me try and satisfy the "quality standards" algorithm by padding it. Yes, I ...
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### A “reverse” diagonal argument?

Cantor's diagonal argument can be used to show that a set $S$ is always smaller than its power set $\wp(S)$. The proof works by showing that no function $f : S \rightarrow \wp(S)$ can be surjective ...
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### What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
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### Fractional cardinalities of sets

Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such ...
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### Cardinal Arithmetic versus Ordinal Arithmetic

I am reading Philosophy, not Set Theory, so please excuse the naivety of my question. My question concerns the wildly different character of ordinal arithmetic versus cardinal arithmetic. The "...
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### Does $\mathcal P ( \mathbb R ) \otimes \mathcal P ( \mathbb R ) = \mathcal P ( \mathbb R \times \mathbb R )$?

Obviously the answer's yes if I replace $\mathbb R$ by a countable set. If $\mathbb R$ is replaced by a set $X$ having a greater cardinality, then the diagonal $\{ (x,x) , \ x\in X\}$ is not in the ...
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### Can proper classes also have cardinality?

In some set theories such as ZF+GAC, in which GAC is global axiom of choice, the Von Neumann universe $V$ bijects to $Ord$, the class of ordinals. It suggests us that proper classes may also have ...
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### Are injections harder to find than surjections?

Given two finite sets $A$ and $B$ with $|A|<|B|$ There are more functions from $B$ to $A$ than from $A$ to $B$ except when $|A|=1$ or $|A|=2,|B|=3,4$. See here for proof. It is also true there are ...
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### Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
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### Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
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### Partition of N into infinite number of infinite disjoint sets? [duplicate]

Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?
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### 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 ...
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### What's the cardinality of all sequences with coefficients in an infinite set?

My motivation for asking this question is that a classmate of mine asked me some kind of question that made me think of this one. I can't recall his exact question because he is kind of messy (both ...
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### Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC

I'm quite sure I'm missing something obvious, but I can't seem to work out the following problem (web search indicates that it has a solution, but I didn't manage to locate one -- hence the ...
### What is the cardinality of a set of all monotonic functions on a segment $[0,1]$?
What is the cardinality of a set of all real monotonic functions on a segment $[0,1]$? Does it really matter that functions are monotonic?