# 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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### Cardinal characteristics — generalisation?

I'm just reading about what is called cardinal characteristics of the continuum. For example there are the bounding number $\mathfrak b$ and the dominating number $\mathfrak d$ etc. which are defined ...
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### Can we conclude that $|\prod_{\xi \in Ord, \xi=1}^{\xi<\alpha}\xi|=2^{|\alpha|}$ for all infinite ordinal $\alpha$ in $\mathsf{ZFC}$?

According to this question, $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa$, the cardinal product of all positive cardinals smaller than a given infinite cardinal $\aleph_\alpha$ ...
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### Can we conclude $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa=2^{\aleph_\alpha}$ in ZFC?

In complex analysis, there is a function called Euler's Gamma function. Whenever given a positive integer $n+1$, it will return $n!=\prod_{i=1}^{i < n+1}i$. I'm not sure if there is similar ...
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### Trouble understanding cardinality

Hi guys I am having trouble understanding cardinality. I am given this practice question. 1) Use Cantor-Schroder-Bernstein Theorem to prove that the intervals $(0,1)$ and $[0,1]$ have the same ...
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### Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
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### minimal infinite sigma algebra [duplicate]

Does there exist sigma algebra whose cardinality is countably infinite? If yes tell me some examples. If not how to show every infinite sigma algebra is uncountable?
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### Is the class of all ordinals independent of set theory?

I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. ...
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### Small claim regarding addition of cardinals

I want to show that if $\alpha,\beta$ are cardinals such that $\alpha=\alpha+\beta$ and $0<\beta$ then $\aleph_{0}\leq\alpha$ It should be fairly simple but for some reason I keep getting stuck.
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### Books/Review material on infinite cardinality for undergrad

You may have noticed me using asking many questions on Infinite Cardinalities on this fine website. Although many of the answers to my questions here were very in-depth and amazing, I just can't help ...
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### If CH assumed, can we prove this?

$$\aleph_2^{\aleph_0}=\aleph_2$$ Appreciate your help
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### If $A$ and $B$ are denumerable sets, and $C$ is a ﬁnite set, then $A \cup B \cup C$ is denumerable

I have a statement here I wish to prove and I would love some help on it :) If $A$ and $B$ are denumerable sets, and $C$ is a ﬁnite set, then $A \cup B \cup C$ is denumerable Here is my thoughts!...
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### About cardinalities of almost disjoint systems of functions

Let us call a system $\{f_i; i\in I\}$ of functions $f_i\colon\omega\to\omega$ almost disjoint if the set $\{n\in\omega; f_i(n)=f_j(n)\}$ is finite whenever $i\ne j$. (I am not entirely sure whether ...
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### Proving every infinite set is a subset of some denumerable set and vice versa

I have 2 sets of statements that I wish to prove and I believe they are very closely related. I can prove one of them and the other I'm not so sure! 1: Every infinite set has a denumerable subset ...
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### “big” Hausdorff space with dense subspace of given cardinality

In a topology course we proved the following proposition: Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a ...
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### Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B$

I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated! Let $A$ be any uncountable set, and let $B$ be a countable ...
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### Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
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### How to prove that a regular cardinal cannot be expressed as a union of sets with less cardinality?

My question is about the following: Using the Axiom of Choice show that: If $\kappa\ge\omega$ is a regular cardinal, $\gamma\le\kappa$, and $\langle A_\alpha\mid\alpha\lt\gamma\rangle$ is a ...
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### If A is a denumerable set, and there exists a surjective function from A to B, then B is denumerable

I am having some trouble solving the following homework question and some help would be greatly appreciated!! Q: Prove that if $A$ is a denumerable set, and there exists a surjective function from $A$...
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### Proving that if $A$ is a non-empty set, then $|A| ≤ |A \times A|$

I just need some help with this problem. Let $A$ be an non-empty set. Prove that $|A| \leq |A \times A|$. $A$ may or may not be infinite! Intuitively, this statement makes sense. $A \times A$ ...
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### Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
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### What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
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### Set theory, property of addition of natural numbers in the cardinal way

Consider the set of natural numbers $\mathbb N$. On this set we define an operation '+', as follows: for all $n,m \in \mathbb N$ we put $n+m$ to be the unique natural number $t \in \mathbb N$ such ...
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(1) $C$ is the set of all circles $C(z,r)$ with $z\in\mathbb{Q}\times\mathbb{Q}$ and $r\in\mathbb{Q}^+$. What is the cardinality of $C$? (2) Let $S$ be the set of all sequences $X=\{X_n\}_{n=1}^\... 1answer 142 views ### Differences in worlds with and without$\aleph_0<|S|<2^{\aleph_0}$Paul Cohen told us that whether or not there is$S$with $$\aleph_0<|S|<2^{\aleph_0}$$ cannot be decided within ZFC, and hence it is reasonable to work in two ... 1answer 105 views ### A question on the power set Let$\mathcal{P}_\kappa(E)$is the collection of all subsets of$E$of cardinality$\le \kappa$, and$[E]^\kappa=\{A: A\subset E, |A|=\kappa\}.$Then$|\mathcal{P}_\kappa(E)|=|E|^\kappa$or only$|\...
I'm trying to prove the following: If it holds that if for any two sets $A$ and $B$, $A$ can be injected into $B$ or $B$ can be injected into $A$, then every set can be well-ordered (axiom of choice ...