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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Proving that 2 intervals have the same cardinality [on hold]

How can I Prove that the invervals [0, 1) and (0, 2] have the same cardinality by finding a bijection between them? And how can I Prove that the intervals (0, 1) and [0, 1] have the same cardinality ...
0
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0answers
17 views

Filter,dual ideal,definition of $\liminf_I \lambda$

We have this http://shelah.logic.at/files/506.pdf definition of $\lim \inf_I \bar{\lambda}$ in 1.1(3): for $I$ a filter on $\kappa$ let $I^+=2^\kappa \setminus I$.$$\lim \inf_I ...
6
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1answer
87 views

For an infinite cardinal $\kappa$, $\aleph_0 \leq 2^{2^\kappa}$

I'm trying to do a past paper question which states: $$ \text{For all infinite cardinals $\kappa$, we have } \aleph_0 \leq 2^{2^\kappa}. $$ I'm supposed to be able to do this without the axiom of ...
1
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1answer
32 views

Saturated models and $\kappa=\kappa^{<\kappa}$

Do not assume GCH. Can you characterize the cardinals $\kappa$ such that every theory $T$ with an infinite model has a saturated model of cardinality $\kappa$? I guess these are the cardinals such ...
1
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2answers
140 views

Is the set of all pairs of real numbers uncountable?

My hypothesis is that $\mathbb{R \times R}$, the set of all pairs $(r_1, r_2)$, of real numbers is uncountable. I understand that the set of all pairs of natural numbers is countable. But could ...
1
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1answer
43 views

Proving $k^{m+l} = k^m k^l$ by constructing a bijective function F : $ ^MK \times ^LK \to ^{L\bigcup M}K $

For cardinals k which is cardinal of K and l which is cardinal of L and m which is cardinal of M. W.T.S [ $ k^{m+l} $ = $ k^m k^l $] by constructing a bijective function F : $ ^MK \times ^LK \to ...
1
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1answer
84 views

Is this behavior of the continuum function consistent with ZFC.

I have an axiom, based on how finite cardinalities work. As we know, every ordinal can be written as the sum of a unique limit ordinal $L$ (where $0$ is a limit ordinal axiom for the purposes of this ...
11
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3answers
728 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
1
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2answers
69 views

How to prove that $C\cdot\aleph_0=C$

How can I prove that $C\cdot\aleph_0=C$? I tried this: Given that $k\cdot 1=k$ and $C\cdot C=C$ if $C\cdot C = C \wedge C\cdot 1 = C \wedge C>|\mathbb N|>1$ then $C\cdot |\mathbb N|= C$ c is ...
2
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0answers
26 views

Existence of infinite set and axiom schema of replacement imply axiom of infinity

I'm self-teaching an intro to set theory course, and came across this exercise: Show that the existence of an infinite set is equivalent to the existence of an inductive set. For the notion of ...
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3answers
33 views

Is the collection of all cardinalities a set or a proper class? [duplicate]

Is the collection of all cardinalities a set or a proper class? Does anybody ever think about the problem?
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0answers
17 views

Cardinality relation between reducible sets

Suppose we are considering natural numbers, set $A$ and $B$ are two subsets of the natural number set, suppose set $A$ is many-one reducible to set $B$, i.e. there is a total computable function $f$ ...
1
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1answer
47 views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
3
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1answer
49 views

Given the axiom of choice, are cardinals ordinals?

Given a model of ZFC, is it correct to talk indistinctly about cardinals and initial ordinals, namely, ordinals $\alpha$ such that for every $\beta < \alpha$, there is no bijection between $\alpha$ ...
1
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3answers
23 views

Cardinality of the set of finite sets in the power set of natural numbers [duplicate]

It is known that $|2^\Bbb{N}|=|\Bbb{R}|$ and that $2^\Bbb{N}$ contains all the subsets of $\Bbb{N}$, just an idea of a question I had and that I would like suggestions on how to tackle. My question ...
2
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1answer
15 views

Can every infinite cardinal $\mu$ such that $\kappa\leq\mu\leq2^\kappa$ be expressed as $\kappa^\lambda$?

Let $\kappa$ be an infinite cardinal. Can I reach every intermediate cardinal $\mu$ with $\kappa \le \mu \le 2^\kappa$ as some power $\kappa^\lambda$? If not, is there another construction that ...
4
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1answer
78 views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
2
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3answers
52 views

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$

Show that if a metric space is complete, separable and not countable then it has cardinal $\aleph_1$ I have encountered this exercise and I don't know where to start. There is a lot of important ...
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3answers
53 views

Are cardinal numbers sets in ZFC?

Are cardinal numbers sets in ZFC, or just proper classes? If they are sets, what is their structure?
3
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1answer
59 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 ...
3
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1answer
31 views

Proving the equivalence of a finite set

Let A be a finite set. Prove that if A≈􏰔n and A≈􏰔m, then n=m. The answer in the book uses a max function, so I was just wondering if there was a simpler way. If not, it would be appreciated if ...
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1answer
31 views

Question about $\text{non}(\mathsf{nonstat}_\gamma)$

A lemma in Kunen's (2011) set theory states that if $\gamma$ is a limit ordinal with $\kappa=\text{cf}(\gamma)>\omega$, then $\text{add}(\mathsf{nonstat}_\gamma)$ $=$ ...
4
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0answers
36 views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that ...
0
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0answers
21 views

finding an injective function to prove cardinality equality

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary ...
5
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1answer
62 views

Cardinality of order topology?

Just out of interest: The cardinality of the Euclidean topology on the real line is $c$. In general, if $X$ is totally ordered of cardinality $\alpha$, the order topology on $X$ must have cardinality ...
4
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1answer
39 views

Is there a simple formula for the cardinality of $\{A\subseteq\kappa\mid |A|\leq\lambda\}$ when $\lambda\leq\kappa$?

If $\lambda\leq\kappa$ are infinite cardinals, how many subsets of $\kappa$ of size $\lambda$ are there? And of size $\leq\lambda$? Is there some sort of explicite formula for this? The internet isn't ...
0
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1answer
39 views

Prove that if $S\subset \mathbb{R}^n$ is not countable, then there exists $x \in S$ such that $x$ is a condensation point.

Let $S \subset \mathbb{R}^n$ with the usual metric. A point $x \in \mathbb{R}^n$ is said to be a condensation point of $S$ if for all $r>0$, $B(x,r)\cap S$ is not countable. Show that if $S$ is ...
0
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2answers
55 views

The number of subsets of cardinality less than $\kappa$ of a cardinal $\kappa$ is $\kappa$

Let $L=\bigcup_{\alpha \in Ord} L_\alpha$ be Godel's constructible universe and thus $L \models GCH$. Let $\kappa$ be an infinite cardinal and $S:=\{A \subseteq \kappa : \#A < \kappa \}$. Is it ...
3
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1answer
44 views

Is this proof that $\kappa^{<\kappa}=\kappa$, when $2^{<\kappa}=\kappa$, correct?

Let $\kappa$ be a cardinal number. I want to show that if $\kappa$ is regular and if $2^{<\kappa} = \kappa$ then $\kappa^{<\kappa}= \kappa$. Here is what I got so far: $$ ...
2
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1answer
65 views

Can we prove AC from the statement “There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$”?

If $X$ is a set, let $\operatorname{CARD}(X)$ denote the Cardinal number of $X$. Let GCH(1) be the statement "If $K$ is an infinite initial ordinal number, then there exists no initial ordinal number ...
6
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4answers
94 views

Is the logarithm of $\aleph_0$ infinite?

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
1
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1answer
26 views

Compare density of rationals to the density of integers

Is is possible to somehow quantitatively compare the density of rational numbers to the density of integer numbers, ascribing to the both a number characterizing the density?
2
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1answer
65 views

equality of Cardinality of $\mathbb{R}$ and $\mathbb{R^2}$

There was a question in our exam which wanted us to prove that $\mathbb{R}$ and $\mathbb{R^2}$ both have same Cardinality. My approach to prove this problem was to try to make a bijection between ...
0
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1answer
22 views

Cardinality of Sets and injections

Let A,B,C,D sets. if |A| $\le$|B| and |B| < |C|, show that |A| < |C| Proof: Case1: suppose |A| < |B| then there exists injection f: A$\to$B and |B| < |C| then there exists injection ...
6
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1answer
148 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
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1answer
44 views

A question about infinite sets and Cantor's Power Set theorem

Let $\operatorname{Card}(X)$ denote the cardinal number of the set $X$. The standard proof of Cantor's Power Set theorem stating that "$\operatorname{Card}(X) < \operatorname{Card}(2^X)$" is ...
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1answer
31 views

Regularity of $\omega_1$ and axiom of choice

Why is the regularity of the ordinal $\omega_1$ a consequence of the axiom of choice?
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3answers
74 views

Can we simplify analysis by getting rid of the uncountable reals? [duplicate]

Since the entire observable part of the universe can only be in a finite number of physically distinguishable states, it seems rather strange that an efficient formal description of the universe would ...
0
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1answer
36 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
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2answers
95 views

Cardinality and Concrete Mathematics

First, let me distinguish Concrete Mathematics ( Mathematic branches that studies fixed structures ) from Abstract Mathematics ( branches that study classes of structures, such as algebra , topology, ...
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1answer
41 views

Set $T$ is Countably Infinite [closed]

How can it be shown that $$T = \{\,(i, j, k) \mid i, j, k \in\mathbb N\,\} $$ is countably infinite?
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0answers
19 views

A question on product space [duplicate]

If $|X|=\mathfrak c$, then what is the cardinality of the product space $X^{\omega}$? Thanks very much.
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3answers
66 views

What properties does $A\to B$ satisfy under 1-1 correspondence?

A 1-1 correspondence between two sets $A$ and $B$ is a function $f\colon A \to B$ satisfying what properties? I do know that we say that two sets $A$ and $B$ are equivalent, and we write $A \sim B$ ...
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3answers
32 views

Limit Ordinals as Infinite Ordinals and other questions

I am studying set theory and I am confused in the following: Are limit ordinals the same as infinite ordinals? I would say yes since the least non-zero limit ordinal is $\omega$. Infinite limit ...
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3answers
83 views

Proving $a + a = a$ if a statement is true

That's the question : Let $a$ be a cardinality such that this following statment is true : For every $A, C$, if $ A \subseteq C$, $|A| = a$ and $|C| > a$, then $|C \setminus A| > |A|$. ...
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2answers
23 views

What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them

As the title suggests, the question is : What is the cardinality of all binary sequnces (infinte and finite) that the sequnce 01 does not apper in them ? I'll tell you where im stuck, let's say f is ...
0
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1answer
23 views

Cardinality and Set Thoery

(|x| = cardinal # of x for clarification) let A,B be two finite sets, show that $|A \cup B| = |A| + |B| -|A\cap B|$ Proof: let $x\in A\cup B$ $x \in (A -A\cap B) + (B- A\cap B) + (A \cap B)$ let ...
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0answers
28 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
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2answers
23 views

Relation between successor cardinals and power sets

What are the known relation between successor cardinals $\kappa^+$ and power sets $2^\kappa$ (when GCH is not assumed)? For example, is it true that $\kappa^+ \le 2^\kappa \le \kappa^{++}$? In ...
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1answer
20 views

Help understanding cardinal multiplication and infinite Cartesian products

The cardinal product of two sets is defined to be the cardinality of the Cartesian product. The Cartesian product is: $$\prod_{\alpha \lt\beta}\kappa_{\alpha}=\{f\mid f\colon\beta\rightarrow ...