All Questions
Tagged with cardinals proof-explanation
40
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Understanding why the diagonal argument for countability fails for an uncountable collection of sets
After reading Rudin's proof, using a diagonal argument, that a union of countable sets is countable, I'm trying to understand why it wouldn't be possible to adapt the argument to an uncountable ...
- 2,248
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Theorem 5, Section 1.4 of Hungerford’s Abstract Algebra
If $K,H,G$ are groups with $K\lt H \lt G$, then $[G:K]=[G:H][H:K]$. If any two of these indices are finite, then so is the third.
Proof: By Corollary 4.3 $G= \bigcup_{i\in I}Ha_i$ with $a_i \in G$, $|...
- 4,249
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Prove that $\mathcal{P}(\mathbb{Z}^+)$ is uncountable
Prove: $\mathcal{P}(\mathbb{Z}^+)$ is uncountable
Here is an excerpt from a proof of this result that I am struggling to follow:
As a part of the proof that $\mathcal{P}(\mathbb{Z}^+)$ is ...
- 402
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Why is cf(α) a cardinal for any limit ordinal α? [duplicate]
In Jech, one of the lemmas state that for every limit ordinal α, cf(α) is a regular cardinal. Every source I tried to search on the Internet claimed that it was obvious to see that cf(α) should be an ...
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Understanding limits with Cardinality
I was reading through a proof (section 16)
Let $\mathbb{Z}_0 = \mathbb{Z}$ \ $\{0\}$ and let $\mathbb{Z}_n = \mathbb{Z}^{n-1} \times \mathbb{Z}_0$ for $n \in \mathbb{N}_+ $ The set $\mathbb{Z}_n$ is ...
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Questions about the induction on cardinals
From Hereditary Cardinality and Rank :
For an infinite cardinal $\kappa$, $$\forall x,\ \textrm{hcard
}x<\kappa\rightarrow\textrm{rank }x<\kappa$$
We can show this by induction on $\kappa$. ...
- 12k
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Prove that if $X$ is finite then $f[X]$ is finite for any function $f:X\rightarrow Y$ and $|f[X]|\le |X|$.
First of all I point out that if $X$ and $Y$ are two sets then the set $Y^X$ is the set of all function from $X$ to $Y$.
So in the text Introduction to Set Theory by Karel Hrbacek and Thomas Jech it ...
- 4,906
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1
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Is the Digit-Matrix in Cantors' Diagonal Argument square-shaped?
It seems to me that the Digit-Matrix (the list of decimal expansions) in Cantor's Diagonal Argument is required to have at least as many columns (decimal places) as rows (listed real numbers), for the ...
- 795
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Proof that a subset of a countable set is countable
I am trying to understand a particular proof that a subset of a countable set is countable. I'm defining "countable" as finite or countably infinite. The exact statement is:
If $S$ is ...
- 2,248
1
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2
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157
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Why $\epsilon \aleph_0 =\epsilon$?
Just a quick question. In the proof of Proposition. 4.14. in Conway's Functional Analysis, it states without any proof that if $E$ is any infinite set with cardinality $\epsilon$ then $\epsilon \...
user200918
1
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1
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153
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Applying Cantor's diagonal argument
I understand how Cantor's diagonal argument can be used to prove that the real numbers are uncountable. But I should be able to use this same argument to prove two additional claims: (1) that there is ...
- 1,062
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uses and proofs of the cardinality of the continuum
after reading this https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
I wonder what good is it as a tool in mathematics if there is no proof. is it merely useful to prove that it is ...
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-2
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561
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How to define a function to show that infinite sets are countable [closed]
To prove cardinality for infinite sets, I know that I have to show that there is a bijection. However, I'm having some trouble defining a function for my sets.
For example: Let O be the set of all odd ...
- 7
2
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2
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127
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How should I interpret this diagram showing the bijection from $(a,b)$ to $\mathbb{R}$
In Chapter 1 of Pugh's Real Mathematical Analysis, Pugh gives the following picture:
I'm aware of other proofs to this like this one: bijection from (a,b) to R
but I'm interested in understanding how ...
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If $\mathfrak{p}\ge5$, then $2^\mathfrak{p}\not\le\mathfrak{p}^2$?
I don't understand some passages of the following demonstration which is present in "The Axiom of Choice" by Thomas Jech.
I don't understand
why $|\mathscr{P}(C_n)|=2^n>n^2$ if we originally ...
- 4,906