Cardinality is a notion of size for sets, usually denoted by $|A|$ as the "cardinality of $A$". With finite sets the cardinality is simply the number of elements which are members of a set. Dealing with infinite sets we can measure them in different ways. Cardinal numbers are very natural in the ...
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Sequence of surjections imply choice
I am reading a paper where a side remark said that if a sequence $\langle g_\beta\colon\omega\to\beta\mid\beta<\omega_1\rangle$ is a sequence of surjections then $\omega_1$ is regular.
I have ...
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251 views
mahlo and hyper-inaccessible cardinals
Wikipedia states that a Mahlo cardinal is hyper-inaccessible, hyper-hyper-inaccessible, etc. Is this a characterisation of Mahlo? If not what about "alpha = hyper^alpha-inaccessible" with the obvious ...
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53 views
$ZFC^- + AFA$ and infinite cardinals
$ZFC^-+AFA$ is a non-well-founded set theory, where $ZFC^-=ZFC-FA$ is $ZFC$ without the axiom of foundation, and $AFA$ is an anti-foundational axiom
With the axiom of foundation we have that every ...
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239 views
Cardinality of sets of functions with well-ordered domain and codomain
I would like to determine the cardinality of the sets specified bellow. Nevertheless, I don't know how to approach or how to start such a proof. Any help will be appreciated.
If $X$ and $Y$ are ...
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644 views
Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)
One of the answers to this question indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality ...
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93 views
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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1answer
84 views
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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102 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 \begin{equation}
\aleph_0<|S|<2^{\aleph_0}
\end{equation} cannot be decided within ZFC, and hence it is reasonable to work in two ...
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93 views
Orders of subgroups of Infinite Profinite Groups
This question is in some sense equivalent to my question here. A proof would answer that question in the case when the base field is perfect.
Let $G$ be a profinite group of cardinality $\kappa$, ...
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Question about the order of a Stationary subset of $ \kappa$
Greets
I'm trying to prove one part of exercise 8.14 of Jech's "Set Theory", namely that if $o(k)\geq k$, then $k$ is weakly inaccessible, where $\kappa$ is regular; $o(\kappa)$ is defined as ...
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77 views
Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$
This is an exercise from Kunen's book.
Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$
are equal.
What I've tried: I want to prove by using induction on $m$. ...
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2answers
112 views
Effective cardinality
Consider $X,Y \subseteq \mathbb{N}$.
We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$.
We say that $X \equiv_c Y$ iff there exist a bijective computable function between ...
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2answers
722 views
Non-existence of a surjection $\aleph_n \to \aleph_{n+1}$, without the axiom of choice
Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ ...
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47 views
A question regarding the Power set
In the proofs that I have seen so far for showing that the power set $2^X$ of a set $X$ cannot be in bijection to $X$, the common idea is to assume that there exists a surjection $f \colon X \to 2^X$ ...
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2answers
71 views
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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99 views
Instance of Continuum Hypothesis implying cardinal inequality
I'm currently trying to solve Exercise 5.27 of Jech's Set Theory (3rd Millennium ed.), viz:
If $2^{\aleph_1}=\aleph_2$, then $\aleph_{\omega}^{\aleph_0} \ne \aleph_{\omega_1}$.
The presumption ...
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Exercise on partially ordered sets from Kunen's *Set Theory*
This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows:
Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
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120 views
Cardinality of Reals and Turing Machines
I'm a math hobbyist, so forgive me if what I ask is silly.
I just learned that the cardinality of Reals is greater than the Naturals.
So, because of that, there can be no turing machines which ...
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1answer
287 views
Finding the cardinality of a set
I have to construct a few things before I get to my question: it's possible we don't need all of it, but I am stuck on the very last step so I should recap everything so far.
Let $\kappa$ be a ...
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Is $2^\infty$ uncountable and is cardinality a continuous function?
I apologize if the title seems too vague, but this is how I was asked the question. So one of my friends intended to write an infinite sum like $\displaystyle \sum_{i=1}^{\infty} a_{2^i}$ .
However, ...
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256 views
cardinality of infinite sets
prove or disprove:
If two infinite sets $A$,$B$ have the same cardinality, then $A\cup B$ and $A$ have the same cardinality.
I even cannot make a judgement.
P.S: Can this be done without using ...
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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 ...
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5answers
202 views
How many cardinals are there?
I'm trying to do the following exercise:
EXERCISE 9(X): Is there a natural end to this process of forming new infinite cardinals? We recommend this exercise instead of counting sheep when you have ...
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Why are infinite cardinals limit ordinals?
My book states this as obvious, but it isn't so trivial to me. thanks
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3answers
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How many $p$-adic numbers are there?
Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where ...
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3answers
301 views
cardinality of set of all real continuous functions
Could somebody explain to me how to prove that the cardinality of all real continuous functions is $c$ ?
The first problem is that I don't know how to show that each real continuous function $f: X ...
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4answers
342 views
What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support?
These questions cropped up in the discussion in this question,
What is the cardinality of the group of bijections from $\Omega$ to $\Omega$ with finite support, where $\Omega=\mathbb{N}$?
...
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2answers
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$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_2\cdot\aleph_1^{\aleph_0}$
I've seen this statement in multiple posts (e.g. here and here), but I can't seem to understand it. I can see why
$$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0},$$
by noting that ...
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130 views
In set theory, what does the symbol $\mathfrak b$ mean?
In set theory, what does the symbol $\mathfrak b$ mean? Could somebody tell me something basic about $\mathfrak b$? In particulat, I want to know the relation between $\mathfrak b$
and $\mathfrak c$.
...
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4answers
225 views
Cardinality of Vitali sets: countably or uncountably infinite?
I am a bit confused about the cardinality of the Vitali sets.
Just a quick background on what I gather about their construction so far:
We divide the real interval $[0,1]$ into an uncountable number ...
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179 views
Cardinal power $\kappa^\kappa$. When is it equal to $2^\kappa$?
Under what assumptions on an infinite cardinal $\kappa$ we have
$$\kappa^\kappa= 2^\kappa?$$
Please delete this question. I know the answer.
4
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1answer
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What is the value of $\aleph_1^{\aleph_0}$?
Is there any neat way to calculate the value of $\aleph_1^{\aleph_0}$?
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3answers
173 views
Looking for Cantor's original proof of the Cantor-Bernstein theorem that relies on the axiom of choice?
Even a sketch of it would be good enough. Thanks.
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Problems about Countability related to Function Spaces
Suppose we have the following sets, and determine whether they are countably infinite or uncountable .
The set of all functions from $\mathbb{N}$ to $\mathbb{N}$.
The set of all non-increasing ...
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174 views
The cardinality of a countable union of sets with less than continuum cardiality
Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this:
$\mathbb R = A_1 + A_2 => |\mathbb R| = ...
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Prove that: $\aleph_0 \cdot \frak{c} = \frak{c} \cdot \frak{c}$
I've been fiddling with this enough.
Found an answer here but didn't quite understand it.
How do I prove that:
$$\aleph_0 \cdot \frak{c} \leq \frak{c} \cdot \frak{c}$$
$$\frak{c} \cdot \frak{c} \leq ...
4
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1answer
64 views
What is the cardinality of $\Bbb{R}^L$?
By $\Bbb{R}^L$, I mean the set that is interpreted as $\Bbb{R}$ in $L$, Godel's constructible universe. For concreteness, and to avoid definitional questions about $\Bbb{R}$, I'm looking at the set ...
4
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3answers
128 views
$\aleph_1\leq A$ for an uncountable set $A$
So we have to prove (axiom of choice is given) that $\aleph_1$ which is the next aleph after $\aleph_0=\omega$ satisfies $$\aleph_1\leq A$$ given that $A$ is uncountable.
Here was my reasoning. The ...
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276 views
Intuition about the size of $\aleph_k$ with $k>1$
Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
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1answer
445 views
Cardinality of a set that consists of all existing cardinalities
I have taken a look at the following topics:
number of infinite sets with different cardinalities
Cardinality of all cardinalities
Are there uncountably infinite orders of infinity?
Types of ...
4
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3answers
301 views
Do the real numbers and the complex numbers have the same cardinality?
So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.
Can the approach be extended to say that the set of complex numbers has ...
4
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1answer
59 views
A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?
A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?
I would appreciate very much an answer
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1answer
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How to show $\kappa^{cf(\kappa)}>\kappa$?
For $\kappa \geq \omega$, which is a cardinal, then how to show $\kappa^{cf(\kappa)}>\kappa$?
My idea: When $\kappa=\aleph_\omega$, then $cf(\kappa)=\omega$, is $\kappa^\omega>\kappa$? It seems ...
4
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2answers
51 views
Question about proof of Hessenberg: $\kappa \cdot \lambda = \lambda$
The following is a theorem:
(Hessenberg) Let $1 \le \kappa \le \lambda$ where $\lambda$ is an infinite cardinal. Then $\kappa \cdot \lambda = \lambda$.
The proof in the book proceeds by transfinite ...
4
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2answers
70 views
About a proof of “$\bigcup A$ is a limit cardinal”
Assume that if $A$ is a set of cardinals such that $A$ contains no largest element and assume that we have shown that $\bigcup A$ is a cardinal. Now we want to show that $\bigcup A$ is a limit ...
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2answers
109 views
how to prove the addition of transfinite cardinal numbers?
How do you prove the following transfinite cardinal addition?:
$ \alpha + \beta = \max(\alpha,\beta)$?
And as the consequence, $\alpha + \alpha = \alpha$ where $\alpha$ and $\beta$ are transfinite ...
4
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2answers
329 views
Number of countable subsets of $\mathbb{R}$
More generally, if a set $S$ has cardinality $\mathfrak{m}$, how many of its subsets have cardinality $\mathfrak{n}$? Clearly there are at least $2^\mathfrak{n}$ such subsets. I don't see how many ...
4
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Fodor's lemma on singular cardinals
Fodor's lemma asserts that if $\kappa$ is a regular and uncountable cardinal, then if $f(\alpha)<\alpha$ for a stationary subset of $\kappa$, then it is constant on stationary subset.
Suppose ...
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If $|X|=|Y|=|X-Y|=\kappa$, can we find a bijection on $X$ that fixes $Y$ only?
in a previous question, I mistakenly attempted to subtract one cardinal number from another. Anyway, this got me to thinking, suppose I have two sets $X$ and $Y$, with $Y\subseteq X$. Suppose also ...
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$\kappa\psi (x,X)\leq \psi (x,X)$
The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point
$x\in X$ is the smallest infinite cardinal number $\tau$ such that
there exists a family $\gamma$
of $\kappa$-sets in $X$ ...
