All Questions
Tagged with cardinals reference-request
39
questions
2
votes
1
answer
64
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Reference request for different definitions of finite
I understand that there are different definitions of finite. They are all equivalent over ZFC, and some of them are even equivalent over ZF, but not over weaker theories than ZF(C). I would like some ...
- 21.4k
0
votes
1
answer
146
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Classification of compact Hausdorff spaces $X$ of cardinality $2^{\aleph_0}>|X|\geq \aleph_1$?
On my own time, I've been curious/investigating some properties of compact Hausdorff spaces of cardinality $\aleph_{\alpha}$, for $\alpha \geq 0$ a particular ordinal. I am aware of the following ...
- 5,830
0
votes
1
answer
68
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Is there a notion similar to cardinality that respects Euclid's axiom?
Is there a notion similar to cardinality that respects Euclid's axiom of "The whole is greater than the part"? I know that a set can have the same cardinality as a proper subset of that set. ...
- 21.4k
0
votes
1
answer
97
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Cardinality of set of Baire functions
I'm reading this paper of Sierpinski. At p.260 he says that it is well known that the set of all injective Baire functions (on the reals) is of cardinality $2^{\aleph_0}$, but he gives no reference. ...
- 4,160
0
votes
1
answer
37
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Definition of $\mathfrak a$-connected graphs?
A typical definition of a graph $G$ being $k$-connected ($k\in\mathbb N_0$) is this:
$k<|G|$ (the order of $G$) and for $X$ being a subset of $V(G)$ such that $|X|<k$ holds $G\setminus X$ is ...
- 3,161
4
votes
1
answer
98
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Simplifying $\left( 2^{\aleph_\alpha}\right)^{\aleph_0}$
Let $X = \omega_\lambda^\omega$ be the product space where each $\omega_\lambda$ has the discrete topology. I'm able to bound the cardinality of open sets in $X$ below by $2^{\aleph_\lambda}$. The ...
- 1,755
1
vote
0
answers
47
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Reference Request: On the proof of $H(x)$ sets in ZF?
Who was the first to prove that for every set $X$, there exists a set of all sets hereditarily strictly subnumerous to $X$, in ZF alone (i.e.; without choice)?
Where $S$ is hereditarily strictly ...
- 4,631
0
votes
0
answers
213
views
Books on infinite sets
I'm a senior undergraduate Pure Math student. I'm looking for books/ problems about large/infinite cardinals and infinite sets. Unsolved or challenging problems are very much welcome.
It shouldn't be ...
- 373
-1
votes
1
answer
211
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Is there a bijection between an infinite set $E$ and $\big\{f:E\to\mathbb{Z}\,\big|\,|\text{supp}f|<\infty\big\}$?
Let $E$ be an infinite set and let $G$ the set of maps from $E$ to $\mathbb{Z}$ that have finite support.
Is there a case where we can prove that there is a bijection between $G$ and $E$?
I need a ...
- 5,193
0
votes
1
answer
69
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Looking for paper with proof by George Cantor
George Cantor proved that the cardinality of $\mathbf{c}$ is larger than the smallest infinity, $\aleph_0$. And he proved that $\mathbf{c}$ equals $2^{\aleph_0}$.
Im looking for the actual paper(s) ...
user366820
0
votes
0
answers
43
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Study materials for cardinals and ω_1 / continuum hypothesis
What textbook (chapter) can I read to understand cardinality concepts like $\omega_1$, $\aleph_1$, and the continuum hypothesis?
The text I'm currently trying to read also uses phrases like "let $\...
- 221
1
vote
0
answers
46
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Reference request about a cardinal related with mad families on $\lambda$ where any two sets in the family meet on fewer than $\kappa$ elements
Let $\kappa$ and $\lambda$ infinite cardinals such that $\kappa\leq\lambda$. We say that two sets $A,B\in\mathcal{P}(\lambda)$ are $\kappa$-$\lambda$-ad iff $|A\cap B|<\kappa$. A family $\mathcal{A}...
- 2,662
3
votes
1
answer
148
views
Consistency of $\mathfrak{b}<\mathfrak{s}$
I'm reading a paper written by Vera Fischer and Juris Steprans related with cardinal invariants of the continuum where they obtain, using finite support iteration of c.c.c partial orders, a model ...
- 2,662
0
votes
0
answers
85
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Generalized Ramsey numbers, possibly infinite?
The Ramsey number $R(m,n)$ is easy to describe. It's the smallest positive integer such that any graph with at least $R(m,n)$ vertices has at least a clique of size $m$ or an independent set of size $...
- 3,161
8
votes
1
answer
289
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Where can I find a proof of ($\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF)?
In "A tutorial on countable ordinals" [1], in page 25, Forster uses the fact that $\aleph_1 \leq 2^{\aleph_0}$ is independent of ZF to prove that there is no definable family of fundamental sequences ...
- 1,457