# 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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### $|X|=|X\cup\{a\}|?$

Let $X$ be an infinite set and $a\notin X$. Prove $$|X|=|X\cup\{a\}|$$ This is so intuitively obvious but upon inspection it appears quite non-obvious. How might one prove this? Do I need the axiom ...
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### Cardinal Inequality without using Choice

Let $k$ be an infinite cardinal (i.e. not in bijection with any natural number), show that $\mathbb{N} \leq 2^{2^k}$ This is very easy with choice, without it I don't even know where to start.
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### Neighborhood chain in $\mathbb{R}^\mathbb{R}$

Edited after SamM's comment: Consider the topological space $\mathbb{R}^\mathbb{R}$, with the usual topology. Pick a point $x \in \mathbb{R}^\mathbb{R}$ and a neighborhood $V = V_0$ of $x$. I wish to ...
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### Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
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### Does $1-\frac{1}{2^\omega}$ equal 1? What about $1-\frac{1}{2^{\aleph_0}}$?

(Correct if I'm wrong on any of this.) Recently, I've been learning about transfinite ordinals and cardinals. For some cases, I understand the difference between ordinals and cardinals, for instance ...
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### Prove that Real Numbers and the interval (-∞,0) have the same cardinality. [closed]

1.Prove that Real Numbers and the interval (-∞,0) have the same cardinality. 2.Prove that Real Numbers and the interval (-∞,0] have the same cardinality. How could I solve this two by using ...
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### Cardinality of the set of $2\times 2$ real matrices with determinant $1$ [closed]

Please, I need the cardinality of the set of $2\times 2$ real matrices with determinant $1$. I have no idea where to start.
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### How to solve probability when sample space is infinite?

I came up with a random problem yesterday: Suppose that in a random trial, each point $(x,y)$ where $x,y \in \mathbb{R}$ and $0 \leq x,y \leq 1$ is assigned a value of $0$ with 50% chance and a ...
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### The generalized continuum hypothesis can't fail first at $\omega_{\omega_{1}}$

I am willing to prove that the GCH cannot first fail at a singular cardinal. For this purpouse I am following the strategy outlined by Kunen in his 2013 book (see Exercises III.6.16-6.17-6.18-6.19). I ...
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### Size of cardinal without choice

How can we show that $\aleph_0 \leq 2^{2^\kappa}$ for any infinite cardinal $\kappa$ without using the Axiom of Choice? By Cantor's Theorem we can easily show that if $\aleph_0 > 2^{2^\kappa}$, ...
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### Checking whether some sets are clubs in $\aleph_2$ and $\aleph_1$

I'm getting acquainted with the stationary sets and clubs. I don't yet quite get everything well, so I'd appreciate some help with this question: which of the following sets are clubs, contain a club,...
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### Connected Linear Graph not Path-Connected

Given a set of vertices $\{x_\alpha\}$ whose cardinality exceeds $\aleph_1$, (assume the axiom of choice) connect each vertex with its successor by an edge, forming a linear graph. Choose two vertices ...
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### Sum of cardinals of all intersections: elegant alternative proofs?

I once read the following problem: compute $$\sum_{A,B\in\mathcal{P}(\Omega)}\operatorname{card}(A\cap B)$$ where $\Omega$ is a set of cardinal $n>0$ and $\mathcal{P}(\Omega)$ the set of the sets ...
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### What is an example of two sets which cannot be compared?

In set theory, if we do not assume the Axiom of Choice, we cannot prove the Trichotomy Law between cardinals. That is, we cannot prove that for any two sets, there exists an injection from one to the ...
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### How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
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### Ultraproduct with no long descending sequence

I have a countably infinite well-ordered structure $M$ (over a countable language if it helps), and an uncountable regular cardinal $κ$, and I wanted to construct an elementarily equivalent structure ...
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### If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$?

If $p_n$ is the $n^{th}$ prime, is it ever appropriate to speak of $p_{\aleph_0}$? I'm no math student. Your pardon if this is just some clearly obvious and easy answer, I'm just not seeing it. ...
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### Well-ordering of sets of cardinal numbers

Proposition For every cardinal number $m$ there is a definite next larger cardinal number. This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal ...
### “Proof” that $\text{cof}(\omega_\lambda)<\omega_\lambda$ if $\lambda$ is a nonzero limit ordinal
The following lemma is from this introduction to cardinals. Lemma 2.7. Let $\omega_\alpha$ be a limit cardinal. Then $\alpha$ is a limit ordinal and $\text{cof}(\omega_\alpha)=\text{cof}(\alpha)$. ...