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.
3,622
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Among 101 dalmatian dogs, each dog has a unique number of black spots, Addition property
Among 101 dalmatian dogs, each dog has a unique number of black spots from the set {1, 2, 3, . . . , 101}. We choose any 52 of the 101 dogs. We want to prove that any set of 52 dogs satisfies the ...
- 343
17
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2
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402
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Combinatorial proof, without axiom of choice, that for any set $A$, there is no surjection from $A^2$ to $3^A$
The well known proof of Cantor's theorem (stating that $A<2^A$ for any set $A$) does not make any use of the axiom of choice. I have now spent some time wondering if the analogous result $A^2<3^...
- 2,243
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2
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Infinite wacky race
Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the ...
- 3,881
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Fixing my gripe with the common proof for showing that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}|$
I am familiar with the proof that shows the powerset of the naturals is of the same cardinality as the reals using binary representation. Here's a quick rundown of the proof:
Showing that $f:(-1, 1) \...
- 377
2
votes
1
answer
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How can I count the number of eventually constant functions $\kappa_1\to\kappa_2$?
Given two infinite cardinals $\kappa_1,\kappa_2$, what's the number $\tau$ of functions $f:\kappa_1\to\kappa_2$ that are eventually constant? I think that, if $\tau_0$ is the number of eventually zero ...
- 310
3
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1
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Is it consistent with ZC that a well-order of type $\omega_\omega$ does not exist?
Working in Zermelo's set theory (with choice for simplicity) - the construction in Hartogs' theorem shows that starting with a set $X$, there is a set $X'$ in at most $\mathcal{P}^4(X)$ (where $\...
- 4,953
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$\{A_i\}, \{B_i\}$ are chain of sets indexed by the same linearly ordered set. If $|A_i|\le |B_i|$ for all $i$, does $|\cup A_i|\le |\cup B_i|$?
Let $I$ be a linearly ordered set and $\{A_i\}, \{B_i\}$ be two collection of sets indexed by $I$,
in an order-preserving way (i.e. $A_i\subsetneqq A_j\iff i<j$ and $B_i\subsetneqq B_j\iff i<j$)....
- 1,932
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Show that $\mathcal{O}$ the set of all open sets in $\mathbb{R}$ has the same cardinality as $\mathbb{R}$
I've seen the post from here Prove that the family of open sets in $\mathbb{R}$
has cardinality equal to $2^{\aleph_0}$
This post is somewhat complex for me, and I turned it to the question as my ...
- 1,364
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what is the probability of sampling a uniform distributed uncountable set and recieving a certain countable subset
I would assume that the chance would be $0%$ as $0=\lim\frac{1}{n}$ but since the equivalent would be something along the lines of $\frac{\aleph_0}{\aleph_1}$ I am not sure.
If $0=\frac{\aleph_0}{\...
- 1
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Are there infinite sets whose aleph and beth numbers are both unknown?
Is there a set $S$ that is definable in ZFC and known to be infinite, but for which we know neither the aleph nor beth number?
For example, we do not know the aleph number of $|\mathbb R|$, but we ...
- 6,672
1
vote
1
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Using Surreal Numbers to measure function growth rate - Tetration?
In "The Book of Numbers" by John H. Conway, pg. 299, he discusses the application of surreal numbers to quantifying the growth rate of functions.
He gives the following correspondences:
$$\...
- 645
1
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$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers?
The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ ...
- 103
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Set theory - functions: Given that $|A|=|B|$, how to prove that $|A^C|=|B^C|$?
Given that $|A|=|B|$ (the cardinality of set $A$ is equal to the cardinality of $B$).
How can I prove that $|A^C|=|B^C|$ (the cardinality of the set of all functions from $C \longrightarrow A$ is ...
- 393
0
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0
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Cofinality of $\beth_\lambda$ and increasing of beth sequence
Premise. I was given the following definition of beth-numbers:
$$ \beth_\alpha := \begin{cases}
\beth_0 = \aleph_0 \\
\beth_{\alpha + 1} = 2^{\beth_\alpha} \\
\beth_{\lambda} = \bigcup_{\alpha < \...
- 509
0
votes
1
answer
90
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Cardinality of Schemes
I was thinking about set theoretic considerations of scheme theory and a question came to me.
I was wondering if there is a way to bound the cardinality of a scheme $S$ (the underlying set of the ...
- 395