# Questions tagged [infinity]

Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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### Infinitely decreasing exponents

Consider the following expression: $$x=a^{{\frac{1}{2}}^{{\frac{1}{4}}^{\frac{1}{8}...}}}$$ Where $a$ can be pretty much anything. Normally, such exponents are evaluated from top to bottom. But this ...
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### Is $|\mathbb{R}^{\infty}| =|\mathbb{R}|$ [duplicate]

Is $|\mathbb{R}^{\infty}| =|\mathbb{R}|$ I started reading Rudin's principle of mathematical analysis and I read this theorem $2.13$ Theorem Let $A$ be a countable set, and let $B_n$ be the set of ...
1 vote
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### The primes have cardinality $\aleph_0$. How does their powerset also have cardinality $\aleph_0$?

Given that the cardinality of the naturals is the smallest infinite cardinal ( $|\mathbb{N}| = \aleph_0$ ), and the primes $\mathbb{P}$ are an infinite subset of the naturals $\mathbb{N}$, we know ...
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1 vote
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### Question regarding validity of argument involving "infinitary construction" of a set

I apologize if the title to this post is not too clear. I was reading Jech's Set Theory and I came across Theorem 2.8: "If $W_1$ and $W_2$ are well-ordered sets, then exactly one of the following ...
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### Proof verification: $P(\mathbb{Z}^+)$ is uncountable

I construct an injection from $(0,1) \setminus \mathbb{Q}$ to $P(\mathbb{Z}^+)$. Here's basically how it works: Take an irrational, say $$0.214560000012511...$$ Split it into strings of digits like ...
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### Minimum sum between permutations of elements in a matrix

So, for the purposes of creating a heuristic for a Sokoban AI, I have to solve the following problem (if you're curious, the elements of the matrix each represent taxicab distances from boxes to ...
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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 ...
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### What is the underlying meaning of LxWxH=Volume? [duplicate]

Say a rectangular solid has length=1, width=3, and height=5. Its base area would be 1x3=3. Its volume would be 5x3. Now, 5x3 means 3+3+3+3+3, and each 3 represents the area of a 2D plane. So does this ...
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### Does an infinite line segment make any sense? [closed]

In school, I was taught that a line segment must be finite. But a fractal has an infinite perimeter. If you just took two points along that perimeter, and "stretched it out" wouldn't you ...
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1 vote
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### Are the "flipped" 10-adic number equivalent to the real segment $[0;1]$ [duplicate]

I just came with an question/idea trying to really feel why real numbers a uncountably infinte. I know Cantor's diagonal argument and I'm quite convinced by it, but here is the idea I had which has to ...
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### Proof of cantor's theorem.

Claim: $P(\mathbb{Z}^+)$ is uncountable. Proof. Suppose its not; then we can let $X_1,X_2,...$ be an enumeration of $P(\mathbb{Z}^+)$. Let $g:P(\mathbb{Z}^+)\setminus\{\mathbb{Z}^+\}\to\mathbb{Z}^+$ ...
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### Hilbert's Grand Hotel is always hosting the same infinite set of guests

I am learning the fundamentals of mathematics. A bit background: This article says that "The mathematical paradox about infinite sets" envisages Hilbert's Grand Hotel: "...a hotel with ...
I have to evaluate the following product: $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}$$ But I couldn't develop anything. I thought about breaking the product into smaller parts with some ...