All Questions
133
questions
1
vote
1
answer
56
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Brun's theorem and the twin prime conjecture
According to the following extract taken from Wikipedia, almost all prime numbers are isolated given Brun's theorem. Doesn't that mean that there is only a finite number of twin prime numbers (they ...
- 37
1
vote
1
answer
116
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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
0
votes
1
answer
45
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Infinite Series of infinite cardinals in ZFC
$\sum_{n=0}^\omega 2^{\aleph_n}=2^{\aleph_\omega}$
Is this true?
And is there a way in ZFC to let $\infty$ range over ALL infinite ordinals (not a concrete one as in the example above) ?
$\sum_{n=0}^\...
1
vote
0
answers
31
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Ordering and dividing orders of infinity.
I read there are an infinite number of orders of infinity.
Can they all be ordered, or are there different orders we can identify where we do not know which has the greater cardinality?
Is the ratio ...
- 21
1
vote
1
answer
109
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Questions about the Infinite Monkey Theorem
(Context: the Infinite Monkey Theorem stipulates that given infinite time, a monkey can type out the complete works of Shakespeare, or any other text of finite length, just by randomly pressing keys.)
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- 467
-1
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1
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75
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Infinite recursion of function defined with respect to "every $n$" [closed]
I have very little formal mathematical training, so I apologize in advance for what may seem like basic issues in this question's phrasing.
I am trying to determine whether I can make a certain ...
- 101
0
votes
2
answers
184
views
Is it true that $\aleph_1=2^{\aleph_0}$, and if so, would the limit of $2^x$ as $x$ approaches $+\infty$ be equal to $\aleph_1$? [duplicate]
I have been reading about cardinal arithmetic in an introduction to set theory and have the following questions that are unclear to me after working through some problems:
How is it that $\aleph_1=2^{...
- 11
4
votes
2
answers
224
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What is the cardinality of the set of injective functions from $\mathbb{N}$ to $\mathcal P \mathbb{N}$?
Question
What is the cardinality of the set of injective functions from $\mathbb{N}$ to $\mathcal P \mathbb{N}$?
Attempt
If we denote the desired set as $I$, then we can find an upper bound by ...
- 4,331
0
votes
1
answer
125
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What is the cardinality of non-singleton subsets of $\mathbb{N}$?
I am studying a course on ZF Set Theory (without the Axiom of Choice) and am currently looking at the cardinalities of infinite sets. One question that I came across is the following:
Determine the ...
- 4,331
1
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0
answers
56
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For what cardinals does exponentiation by a smaller cardinal is idempotent.
It is true that $\mathfrak{c} = \mathfrak{c}^{\aleph_0}$ but $\mathfrak{c} < \mathfrak{c}^\mathfrak{c}$.
Is it true that $\mathfrak{c} = \mathfrak{c}^{X}$ for all $X < \mathfrak{c}$?
The ...
- 113
0
votes
0
answers
57
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Reaching the continuum through countably infinite steps
Let the following set,
$$S_n = \left \{0, \frac 1 n, \frac 2 n, \dots, \frac {n-1}{n}\right\}, \quad n \in \mathbb N_+.$$
I read that if
$$n\to \infty \implies S_n \to [0,1]. \tag{1}$$
Namely, that as ...
- 10.4k
3
votes
0
answers
153
views
Is the infinite product of {0, 1} countable?
In my math class, we had an exercise asking us to prove that the following set is not countable:
$$\prod \limits^\infty_{i=1} \{0,1\} = \{0,1\} \times \{0,1\} \times \{0,1\} \times \cdots$$
By Cantor'...
- 141
0
votes
1
answer
80
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Mathematical implications of an infinite cosmos - distance between two arbitrary points?
On the surface, this may seem like physics or cosmology, but I suspect my question has more to do with a misapplication of math. If it seems like a bad fit here, yell at me and I'll move it.
Let us ...
- 6,014
0
votes
1
answer
93
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Help understanding countable and uncountable infinities
just had some questions about countable and uncountable infinities.
If we take a limit that results in $\frac{ \infty }{0}$, we typically conclude that the limit is just $\infty$, correct? But if the ...
- 515
2
votes
1
answer
170
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Preimage of zero under a continuous function on compact real interval has at most countable connected components
As part of a larger inquiry, I suspect and am trying to prove the following :
Let $\phi$ be a continuous function defined on $[0,1]$. Then $\phi^{-1}(0)$ has an at most countable number of connected ...
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