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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1answer
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Proving $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$ for $X=\mathbb{R}^{\mathbb{R}}$

Let $X=\mathbb{R}^{\mathbb{R}}$. Claim: $|X^{\mathbb{R}}| \lt |\mathbb{R}^X|$ How can this be shown? Note: We are assuming AC and $A^B$ represents the set of functions from B into A
1
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1answer
51 views

Comparing cardinalities

Why these two sets are equinumerous? $$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$ Here is my reason: The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and ...
1
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1answer
66 views

Cartesian product of large sets

For a non-empty set $A$ let $A'$ denote the Cartesian product of $A$ with itself taken denumerably many times. Now given a set $S$ whose cardinality is strictly greater than the cardinality of ...
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1answer
27 views

A question about an exercise on basic cardinal arithmetic.

I just want to make sure that I have proved the following exercise correctly. Given two cardinal numbers $a$ and $b$ where $a$ is infinite. I was to show that $2\le b \le 2^a \implies b^a=2^a$ I ...
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1answer
38 views

Cardinality of a set of closed intervals

What is the cardinality of the set S of all closed intervals on the real number line with rational (positive) lengths? I believe the number of intervals with a specific fixed length but varying start ...
0
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1answer
36 views

dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?
11
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0answers
400 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 ...
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0answers
132 views

Sets of Cardinals without choice

I have a theory that the axiom of choice is equivalent to the statement that sets of distinct cardinals are well ordered by cardinality. I can prove that the axiom of choice implies this. However I am ...
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0answers
40 views

There's no cardinal $\kappa$ such that $2^\kappa = \aleph_0$

I am trying to prove that a cardinal $\kappa$ such that $2^\kappa = \aleph_0$ . My attempt: We suppose it exists. Since $\kappa<2^\kappa$, in particular, $\kappa<\aleph_0$. But that implies ...
6
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0answers
230 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?
6
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0answers
130 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 ...
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0answers
64 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 ...
5
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0answers
175 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 Hopkinsā€“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 ...
4
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0answers
54 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
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0answers
58 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 ...
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0answers
39 views

Can every cardinal number between $\kappa^+$ and $2^\kappa$ be realized in this way?

(Assume ZFC for the entire question. By a tree, I mean a tree in the sense of set theory. I write $h(T)$ for the height of a tree $T$, and $h(f)$ for the height of an element $f \in T$.) Definition ...
4
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0answers
75 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 ...
4
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0answers
86 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
77 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 $\{H_\alpha: \alpha < \kappa\}$ ...
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0answers
39 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 ...
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0answers
175 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 ...
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0answers
46 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 ...
3
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0answers
56 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 ...
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0answers
97 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
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0answers
67 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
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0answers
114 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
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0answers
45 views

Weight of topology related to sizes of open sets?

I'm wondering if there is a notion relating the weight of a topological space (=minimal base cardinality) to the sizes of its open sets. In particular I'm looking for properties of spaces, whose ...
2
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0answers
90 views

How large is an uncountable regular cardinal which is closed under arbitrary fast operators?

Let $Card$ be the proper class of all cardinals, define an infinite set of operators like $\otimes_{n}:(Card\setminus \omega)\times (Card\setminus\{0\})\longrightarrow Card$ which are defined for each ...
2
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0answers
39 views

What is exactly meant by a countable collection in the defintion 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
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0answers
99 views

Why are large cardinal axioms actually axioms?

A cardinal is an isomorphism class in ZFC, or a representative of one. I'm not asking what the significant uses of large cardinals are, or why we would want to find them or construct them; I'm ...
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0answers
24 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 ...
2
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0answers
52 views

A question on semi-stratifiable spaces

A space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: (i) for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ...
2
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0answers
77 views

A quick question about proof of Bukovský-Hechler

The following is an exercise in Just/Weese (page 179), Question 1: can you tell me if I got it right? Thank you! Question 2: Shouldn't it be equality rather than less equals in $\mu = \sum_{\alpha ...
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0answers
40 views

Existence(?) of a set whose cardinality cannot be determined in ZFC

(First, I apologize if I display any fundamental misunderstanding of how set theory works.) I had a question regarding the limitations of ZFC (assuming its consistency, of course.) Is there any ...
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0answers
54 views

Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
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0answers
48 views

Characterizability in $L^2_{\kappa^+\omega}$

I'm reading an article on second order characterizability. At some point in the article it proves that any model $A$ of cardinality $\kappa$ must be characterizable in $L^2_{\kappa^+\omega}$. I.e. ...
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0answers
38 views

Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional banach space $X$?

So given an arbitrary infinite dimensional Banach space $X$ can we deduce that it's dimension $\dim(X)$ (the cardinality of one of it's Hamel bases) is less or equal of the dimension $\dim(X^{\ast})$ ...
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0answers
33 views

Decomposition of an infinite set into pairwise disjoint subsets which exhaust set does not affect cardinality

Show that if $X$ is an infinite set and $A$ is subset of the power set of $X$ containing only finite pairwise disjoint sets such that the union of all elements of $A$ is $X$, then cardinalities of $X$ ...
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0answers
64 views

stationary subset of $cf(\kappa)$

Let $\kappa$ a infinite cardinal with uncountable cofinality and $S\subset\kappa$. We can find a normal function $f$ from $\operatorname{cf}(\kappa)$ on $\kappa$ such that ...
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0answers
83 views

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory ...
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0answers
22 views

$\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$

$\mathbb R$ is the set of the real numbers. $\mathbb Q$ the set of the rational numbers. So how I can prove this? $\mathbb R$ ~ $[0,1) \cap (\mathbb R\ $\ $\mathbb Q)$ I am also not sure what means ...
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0answers
36 views

Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
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0answers
46 views

Cardinal Arithmetic Example Wikipedia

Hello I am studying cardinal arithmetic, and found out that I found that $\mathfrak{c}^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \aleph_0} = 2^{\aleph_0} = \mathfrak{c} $. However I found ...
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0answers
90 views

Existence of a Banach space with algebraic dimension $\alpha \neq \aleph_0$

For any given cardinal number $\alpha \neq \aleph_0$, is there a Banach space $X$ with algebraic dimension $\alpha$?
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0answers
68 views

Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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0answers
23 views

Is 2 a regular cardinal?

there are different definitions of regular cardinals. (1)a cardinal k is regular if cf(k)=k,since 2 is a successor cardinal,cf(2)=1.so cf(2) is not 2,so 2 is not regular. (2)a cardinal k is regular if ...
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0answers
29 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
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0answers
33 views

cardinality of finite subset

I have the following question: Prove that a set A has the same cardinality of a subset of a Set B, if and only if exists an injective function A to B. I find it hard to prove it because I can easily ...
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0answers
54 views

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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0answers
193 views

Enumeration of subsets

I want to show that if $\{X_\beta : \beta<\lambda^+\}$ is an enumeration of $[\lambda\times\lambda^+]^{\leq\lambda}$ then, for all $i<\lambda$, the set $\{(X_\beta)_i:\beta<\lambda^+\}$ is ...