# 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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### Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
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### Finding an example of a bijection from $\Bbb N$ to $E^+$.

Give an example of a bijection $h$ from $\Bbb N$ to $E^+$ such that $h(1) = 16, h(2) =12, \text{ and } h(3) = 2.$ $\Bbb N = \text{ natural numbers }$ , $E^+= \text{ positive even integers. }$ So ...
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### Is the set of aleph numbers countable?

If I write the set of aleph numbers in this way $\{\aleph_0, \aleph_1, \aleph_2, \aleph_3, \dots\}$ it seems obvious to me that this set is countable, because aleph numbers have integer coefficients. ...
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### Without AC is there a relationship between $\beth$ and $\aleph$ numbers?

Assuming AC we know that all $\beth_\alpha$'s will be $\aleph_\beta$ for some $\beta$ since they can be well ordered. Can anything interesting be said about their relationship without AC? Is it ...
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### Enderton's Elements of Set Theory Scott's Trick Exercise (page 207 problem 31)

31. Define kard $A$ to be the collection of all sets $B$ such that (i) $A$ is equinumerous to $B$, and (ii) nothing of rank less than rank $B$ is equinumerous to $B$. (a) Show that kard $A$ is a ...
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### How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
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### An easy to understand definition of $\omega_1$?

I have two things I'm not sure in 100% about them. The first, is $\omega_1$. I have a little "feeling" of it, but if I'll be asked to define it - I don't know where to begin from. Perhaps it is ...
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### Show that a set with an uncountable subset is itself uncountable.

Let $A = P \cup Q$, where $P, Q$ are disjoint [1] and $P \ne \emptyset$ is countable and $Q \ne \emptyset$ is uncountable. Then $Q \subset A$ [2]. Show that $A$ is uncountable. Proof (by ...
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### Did Cantor both prove and disprove the Continuum hypothesis?

I have been listening to the podcasts of A Brief History of Mathematics on BBC Radio 4. In the episode on Georg Cantor the narrator, Prof. Marcus du Sautoy, says that one day Cantor proved that there ...
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### If a set $A$ is uncountable , and a set $B$ is countable then $A \times B$ is uncountable.

I prove it by contradiction. Let $A \times B$ is countable. It means we can list down the all the ordered pairs of $A \times B$. So if ordered pairs of the form $(a,b)$ are countable (where $a \in A$ ...
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### Lowering the cardinality of a set?

Given a set X with a certain cardinality, there are explicit constructions for getting a set with the "next bigger" cardinality, e.g. constructing the power set. Does some analogous construction ...
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### prove sets cardinality inequality

I need to prove that if $$A , B$$ are infinite sets and it holds that : $$|A| > |B|$$ then: $$|A \backslash B| = |A|$$ I guess I just don't what can I say about the cardinality of |A\B| ...
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### Cardinality question on set of symbols [closed]

Few moments ago I asked myself a question, that I not positive if, in fact, is well defined. Let $\mathbb{R}$ be the set of real numbers. Define $S$ to be a set of symbols, as follows: Let $x$ be ...
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### Does there exist a bijection from $[0,1]$ to $\mathbb R$?

We can find a bijection from $(0,1)$ to $\mathbb R$. For example, we can use $f(x)=\frac{2x-1}{1+|2x-1|}$ composed of parts of two hyperbolas, see the graph here. Or we could appropriately scale the ...
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### Proving $x\preceq y \implies \bar{\bar{x}} \leq \bar{\bar{y}}$ in cardinal arithmetic

Let $\bar{\bar{x}}$ denote the cardinal of $x$ and $\approx$ denote bijective equivalence. Assume $x\preceq y$. By definition $\exists z (z \subseteq y \land x \approx z)$. Now from something I've ...
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### Proving a set is Uncountable or Countable Using Cantor's Diagonalization Proof Method

I understand the idea that some infinities are "bigger" than other infinities. The example I understand is that all real numbers between 0 and 1 would not be able to "fit" on an infinite list. I have ...
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### Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
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### Let $R$ be an infinite comutative ring with unity, $M,N$ be $R$-modules, $f:M \to N$ be a surjective module homomorphism; then $|M|=|N ||\ker f|$?

Let $R$ be an infinite commutative ring with unity, $M,N$ be modules over $R$, let $f:M \to N$ be a surjective module homomorphism; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
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### Counting subsets of different sizes of a set

Let $A$ be a non-empty set, and $\mathcal{P}^*(A)$ denote the power set of $A$ excluding empty set. There is a natural equivalence relation on $\mathcal{P}^*(A)$: for $S_1,S_2\in \mathcal{P}^*(A)$...
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### Paths in a connected subset of $\mathbb R^n$

Prove that every connected subset $X$ of $\mathbb R^n$ with more than one point has the continuum cardinality. In order to use Baire Lemma, I would like to demonstrate that there is a compact (or ...
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### Existence of model of ZF in which every uncountable cardinal is the cardinality of some power set?

Does there exist a model of ZF in which any uncountable cardinal number is equal to the cardinality of the power set of some set ? ( In ZFC it is not possible as is shown by the answers to this ...
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### Properties of the power set of $A$

Let $A$ be any set . Let $\wp(A)$ be the power set of $A$. Then which of the following are true 1) $\wp(A) = \emptyset$ for some $A$ 2) $\wp(A)$ is a finite set for some $A$ 3) $\wp(A)$ is a ...
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### Are there any constructive axioms which disprove the continuum hypothesis?

I understand that the Continuum hypothesis is independent of ZFC, so that we may comfortably add either the continuum hypothesis or its negation to ZFC without creating any paradoxes (unless ZFC had ...
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### How can I remember whether finite or countable cartesian product of countable set is countable

I always forget this result Is cartesian product of countable set countable under finite or countable cartesian products? Is there a good way to remember this? Like a proof sketch where the ...
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### Good introduction to cardinals?

is there a good text book to cardinals? I am more interested in how cardinal works than cardinality. Because it seems in undergrad they cut off at proof of $\mathbb{R}$ is uncountable and does not go ...
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### If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

Claim: If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$ $\coprod$ is the disjoint union of disjoint sets $A_n \subset A, \forall n \in \mathbb{N}$ Is ...
### Bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$
How can we construct a bijection from $\mathcal{P}(\mathbb{N})$ to $(0,1)$? Here is what I know: $\mathcal{P}(\mathbb{N}) = \{A | A \text{ is a subset of } \mathbb{N}\}$ Both \$\mathcal{P}(\...