Questions tagged [cardinals]

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.

16
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667 views

Formulations of Singular Cardinals Hypothesis

I have stumbled on a few different formulations of the Singular Cardinals Hypothesis. The most common are: SCH1: $\quad 2^{cf(\kappa)}<\kappa \ \Longrightarrow \ \kappa^{cf(\kappa)}=\kappa^+$ for ...
12
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0answers
312 views

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 ...
9
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0answers
207 views

Can such a Dedekind cardinal exist?

Motivated by idle curiosity and this question about characterizing countable sets I ask: Is it consistent with ZF that there is an uncountable set $S$ such that, for every infinite set $X\subseteq ...
9
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0answers
256 views

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 ...
9
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0answers
184 views

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 ...
8
votes
0answers
270 views

About singular $\beth_{\alpha}$ for limit ordinals $\alpha$

Why are there cofinitely many regular cardinals of type $\beth_{\beta}$ below any given singular $\beth_{\alpha}$, if $\alpha$ is a limit ordinal?
7
votes
0answers
230 views

Cardinality of Galois groups

We know that no Galois group of a Galois extension is countable. The question is: which cardinalities are possible for a Galois group? (or also: for profinite groups?) I suspect that the theory of ...
7
votes
0answers
228 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
6
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0answers
131 views

Two well orderings of an infinite cardinal agree on a large set

I've seen this question but I'm having trouble following the proof given. This is an exercise from Kunen: If $\kappa$ is an infinite cardinal and $\triangleleft$ is a well ordering on $\kappa$, then ...
6
votes
0answers
245 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that $\aleph_n^{\...
6
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0answers
150 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without unity)....
5
votes
0answers
75 views

A question on algebraically closed field

Let $k$ be an algebraically closed field . Consider the commutative ring , with unity , $A=k^\mathbb N=\prod_{i\in \mathbb N}k$ . Consider the proper ideal $I=\oplus_{i\in \mathbb N}k(=k^{(\mathbb N)})...
5
votes
0answers
122 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} \...
4
votes
0answers
51 views

Cardinality set of downsets

Let $(X, \leq)$ be a totally ordered set. A downset of $X$ is a subset $A$ of $X$ with the property $$ \forall x \in X \forall a \in A : ( x \leq a \Rightarrow x \in A ) $$ Denote $\mathcal{D}(X)$ for ...
4
votes
0answers
164 views

Why is the weight of a topological space a minimum?

If $(X, \tau_X)$ is a topological space, then the weight is usually defined as follows: $$w(X) = \min \{ \vert B \vert : B \subset \wp(X), B \mathrm{\; is \; a \; basis \; of \;} \tau_X \}$$ I was ...
4
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0answers
1k views

Is it possible to create division via Set Theory?

I've been reading a book on Set Theory (Charles C. Pinter), and it says, ...set theory is recognized to be the cornerstone of the "new" mathematics... [emph. added] and that ...we can still ...
4
votes
0answers
113 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S :=...
4
votes
0answers
310 views

Solovay Theorem

Solovay's Theorem states If $\kappa$ is regular uncountable, then any stationary set in $\kappa$ can be partitioned into $\kappa$-many pairwise disjoint stationary sets. For regular uncountable ...
4
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0answers
100 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
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0answers
70 views

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 ...
4
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0answers
198 views

Property of Galvin-Hajnal rank

In what follows, $\|\Phi\|_I$ means the Galvin-Hajnal rank of a function $\Phi:\kappa \to Ord$ with respect to a $\omega_1$-complete ideal $I$ of $\kappa$. Lemma 2.2.5 of Introduction to Cardinal ...
3
votes
0answers
47 views

Equivalent finite sets have the same number of elements (odd proof)

I am working through Rudin's POMA and in Chapter 2 on Basic Topology he gives the following definition of one-to-one correspondence: If there exists a 1-1 mapping of $A$ onto $B$, we say that $A$ ...
3
votes
0answers
98 views

Is there an alternative characterisation of countably closed cardinals?

Recall that a cardinal $\kappa$ is countably closed if $\nu^{\aleph_0}<\kappa$ for every $\nu<\kappa$. I wanted to get some kind of an intuition of countably closed cardinals as well as ...
3
votes
0answers
113 views

How many non-isomorphic groups of infinite order $|X|$ are there?

Let $X$ be an infinite set. What I'm looking for is a lower bound (or even better, precise cardinal number) of possible non-isomorphic group structures on $X$. Since every group is built on some ...
3
votes
0answers
446 views

Are there any theories that deny the continuum hypothesis that are used outside metamathematics?

ZFC is consistient with the negation of the continuum hypothesis. Do theories that deny the continuum hypothesis have any use outside of metamathematics at the moment? For example, I know that ...
3
votes
0answers
111 views

Minimal Possible Ordinal for a Cardinal

For a given $\aleph_\alpha$, what is the minimal ordinal possible such that $|\beta| = \aleph_\alpha$? More precisely, assume we have a model $N$ of ZFC. $N$ could be an extension of many different ...
3
votes
0answers
53 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
3
votes
0answers
110 views

Proof check:$ \left | \mathbb{R} \right |= 2^{\left|\mathbb{N} \right |}$

This is my first time to post here. Sorry if this post is too simple or naive. Here I would like to prove that $\left | \mathbb{R} \right |= 2^{\left |\mathbb{N} \right |}$ I would first ...
3
votes
0answers
140 views

Why is this not a proof of Schroeder-Bernstein?

We can show that if $f: A \rightarrow B$ is injective then $|A| \leq |B|$ and if $g: B \rightarrow A$ is injective then $|B| \leq |A|$ so $|A| = |B|$. By the definition of having equal cardinality, ...
3
votes
0answers
124 views

A problem with an assumption in a previous lemma for the proof of Silver's Theorem on SCH in Jech's “Set Theory”

In the Jech's textbook proof of Silvers Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof. First, the Lemma 8.15 says that, under the assumption that $\...
3
votes
0answers
179 views

$\mathfrak b \leq \mathfrak r$. Is this proof correct?

I am trying to prove that $\mathfrak b \leq \mathfrak r$ where $\mathfrak r$ is the minimal cardinality of a reaping family of sets. A family $\mathcal R \subseteq [\omega]^\omega$ of sets $\mathcal ...
3
votes
0answers
83 views

Which Field Would You Use to Represent a Group Larger than $\aleph _1$?

I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex ...
3
votes
0answers
216 views

Can any large cardinal axiom be reached by a recursive sequence of unbounded sequences and fixed points?

The lowest large cardinals can be seen as a simple recursive rule that defines an unbounded sequence, then defines a fixed point of the sequence, then an unbounded sequence of fixed points, a fixed ...
2
votes
0answers
41 views

Functions between ordinals.

I'm trying to compute the cardinality of a determined set. $$A=\{f\colon \omega_n \to \omega_m | |Supp(f)|=\aleph_k \quad k<n\}$$ As suggested by the exercise, I first tried few elementar cases: $$...
2
votes
0answers
63 views

Hypertask, Arithmetical hierarchy and beyond

Good day, I would love to ask this question. Lets have a hypercomputer capable of doing a hypertask, that is performing uncountably many computational steps in finite time(the same amount of steps ...
2
votes
0answers
129 views

Where can I find a proof of Shelah mentioned in the book “Almost Free Modules” as Theorem VI.5.8?

The following is stated as Theorem VI.5.8 in the newer edition of the book "Almost Free Modules" by Eklof and Mekler, but no proof is given and no precise reference is given. Presumably it comes from ...
2
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0answers
85 views

Determine whether or not $card(A^B)>card(C^D)$?

Let's say I have $4$ sets: $A,B,C,D$, is there a way to determine whether or not $\left|A^B\right|>\left|C^D\right|$? I thought about this question because it is easy to show that $\left|n^\Bbb N\...
2
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0answers
46 views

Cardinal Arithmetic - Right or Wrong?

${\,}| {\Bbb{R}} {\rightarrow} {\Bbb{N}} {\,}| = {\,}| {\Bbb{N}}^{\Bbb{R}} {\,}| = \aleph_0^{\aleph} = \aleph_0{^2}^{\aleph_0}$ ${\,} | {\Bbb{R}} {\times} ({\Bbb{R}} \rightarrow \{0,1\}) {\,}| = ...
2
votes
0answers
161 views

Bijection between the set of all polynomials and $\mathbb{R}^\mathbb{N}$

A question I have been given asks me to determine whether there is a bijection between the set of all real polynomials P and the set of (potentially infinite) real sequences $\mathbb{R}^\mathbb{N}$. ...
2
votes
0answers
83 views

Cardinality of a Banach space

Let $X$ be a separable infinite-dimensional Banach space. Assume that every element of $B_{X^{**}}$ is the weak-star limit of a sequence in $B_{X}$. Clearly, by the canonical embedding, we can see ...
2
votes
0answers
125 views

Give applications of $\beth_3$ other than $\beth_3=\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))$

What are a few applications of $\beth_3$ other than $\beth_3=\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathbb{N})))$ or any other variations of generalized identities that can easily be found on Wikipedia? ...
2
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0answers
84 views

If $A\approx B$ , $C\approx D$ and $A \cap C \approx B \cap D$ is it true that $A \cup C \approx B \cup D$?

I have noted that when $\lvert A\cap C\rvert = \lvert B\cap C \lvert = n$ for a finite $n$ number, the proposition holds, but if the intersection is infinite, I am not sure. I know that there are ...
2
votes
0answers
171 views

uncountable union of uncountable sets and the usage of the axiom of choice

I've been facing the next problem: $\alpha$ is an ordinal, show that there is no cardinals collection $\{\beta_i\}$, when $i\in I$. Such as: $\sum_I \beta_i=\aleph_{\alpha+1}$. When $|I|<\aleph_{\...
2
votes
0answers
35 views

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 ...
2
votes
0answers
44 views

Equal sets have power sets of equal order?

If two sets say $S$ and $T$ are equal is it true that $|2^{S}| =|2^{T}|$. Here is the motivation. Suppose that $S$ has infinite (or countable) order but that is is written as the union of a finite ...
2
votes
0answers
209 views

weight of a topological space

Let $\omega(X)=\min\{|\mathcal{B}|:\mathcal{B} \mbox{ is a base of the topology of } X\}$. In https://en.wikipedia.org/wiki/Base_(topology), it stated that if $\mathcal{B}$ is a basis of $X$, there is ...
2
votes
0answers
106 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
2
votes
0answers
59 views

Clarification on the proof of Theorem 8. 11 (Hungerford)

If $\alpha$ and $\beta$ are cardinal numbers such that $0\neq \beta \leq \alpha$ and $\alpha$ is infinite, then $\alpha\beta=\alpha.$ Sketch: Let $A$ be an infinite set with $|A|=\alpha$ and let $\...
2
votes
0answers
104 views

What is exactly meant by a countable collection in the definition of a sigma-algebra

This a just a small question about the definition of a sigma-algebra and what is eaxactly meant by countable? Would be grateful for any clarification on this. Most texts define a sigma-algebra ($\...
2
votes
0answers
29 views

(S is a proper class wrt model M) implies ((|T|=|S| under model N) implies (T is a proper class wrt M)). Why?

If I understand correctly (which is far from guaranteed, so a reply telling me that it is rubbish will be better than no reply at all): Suppose we have two models N and M such that N is a (non-...