All Questions
Tagged with cardinals real-numbers
61
questions
0
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2
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57
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Are there any theorems that use the uncountability of the reals in their proof?
Can we use the uncountability of the reals as a tool to prove any theorem? Can we use this to calculate anything? Suppose I was trying to convince a pragmatist that uncountability is useful.
- 470
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1
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115
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Shouldn't ℵ₀ be the cardinality of the reals?
If in ZFC any set can be well ordered, and that $\aleph_0$ is the cardinality of every infinite set that can be well ordered, shouldn't $\aleph_0$ be the cardinality of the real numbers?
I know this ...
1
vote
1
answer
148
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Why isnt $|\mathbb{R}| = |\mathbb{N}|$?
Question: To show that 2 sets have the same cardinality, there needs to be atleast one bijective mapping between them. So given the below proof of a bijective mapping below, why can't we say that $\...
-1
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2
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128
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The rationals and their initial segments, edited version [closed]
This post was inspired by an Alon Amit post on Quora. The Quora
problem posed to AA was something like, only slightly more confused
than, this: How can the set of initial segments of the rational ...
- 23
0
votes
1
answer
84
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VC dimension and Cardinality of real numbers
Is it true that the cardinality of a hypothesis class with finite VC dimension is less than the cardinality of real numbers?
My intuition is that the number of functions in a hypothesis class with ...
- 991
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1
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33
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Uncountable union of chain of subsets of R
For each $t \in R$, let $E_t$ be a subset of $R$. Suppose that if $s<t$ then $E_s$ is a proper subset of $E_t$.
$$\bigcup_{t\in R} E_t$$
Is countable.
How?
I see that the union runs over R which is ...
- 389
2
votes
2
answers
118
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Bijective function between $\mathbb{R}^n$ and $\mathbb{R}^m$
I suspected that it was not possible to define a bijection $f \colon \mathbb{R}^m \to \mathbb{R}^n$ where $m,n \in \mathbb{N}$ and $m>n$.
After coming across Why are the cardinality of $\mathbb{R^...
- 77
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2
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73
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We can show bijection between [a,b] and [c,d] in $\mathbb(R)$. But how can one establish an bijection between $[a,b] and (c,d)$. [duplicate]
We can show bijection between [a,b] and [c,d] in $\mathbb(R)$ by send $x \in [a,b]$ to $(d-b)/(c-a)x + bc-da$. But how can one establish an bijection between $[a,b]$ and $(c,d)$ can we explicitly ...
2
votes
2
answers
393
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Any two sets of continuum cardinality are equinumerous?
Is it true that any two uncountable subsets of $\mathbb{R}$ is equinumerous? If yes how to prove that? More generally can I say two sets of same Cardinality are equinumerous?
equinumerous means there ...
- 632
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1
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1k
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Show that $\mathbb{R}$ and the open interval $(-1, 1)$ have the same cardinality.
I am a little confused about using functions to show that two sets of intervals have the same cardinality.
I believe that if we can find a bijective function $f$ such that $f: \mathbb{R} \to (-1, 1)$, ...
- 115
1
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0
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56
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How to linearly order the set of all subsets of real numbers?
I wondered if there are linearly ordered sets of any cardinality. As I understand it, there are. But I want to see at least one concrete example of a linearly ordered set which cardinality is greater ...
- 21
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votes
1
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61
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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
2
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1
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250
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Is almost every definable number uncomputable?
We know that almost all real numbers are undefinable.
We also know that almost all real numbers are uncomputable.
We also know that there are numbers that can be defined but not computed.
However, ...
4
votes
1
answer
79
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Modifying proof of uncountability
My professor's lecture notes offer a proof that $\mathbb{R}$ is uncountable. I think it is slightly incomplete, but I just want to be sure.
No map $f: \mathbb{N} \to \mathbb{R}$ can be surjective ...
7
votes
1
answer
183
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Proof that $\mathbb{R}$ is not countable
I know that this proof may sound ridiculous, but I'm really curious to find out if it's logically correct(and whether there are some circularities). Since $|[0,1]|=|\mathbb{R}|$, we have to simply ...
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