3
votes
2answers
84 views

Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$?

As the title says, my question is: Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$? I'm fairly certain this is true for finite sets but maybe ...
0
votes
1answer
35 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
1
vote
1answer
38 views

Cardinality of Orderings of $\mathbb{R}$

For a finite set $S$ there are $\vert S\vert!$ orderings of its elements. What is the cardinality of all orderings of $\mathbb{N}$? What would $$\vert \mathbb{N}\vert!$$ mean? Is it ...
0
votes
0answers
23 views

Infinite One-Time Pad

As you know, when used correctly, a one-time pad allows one to send a message, such that the only thing that can be found out about it is the maximum size (which is also the key length.) It is ...
0
votes
1answer
90 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
2
votes
1answer
96 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
3
votes
4answers
206 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
1
vote
1answer
40 views

Show A is countable infinity

One more question about set theory: $A\subseteq R$ is an infinite set of positive numbers. Assume there is a value $k \in Z$ such that for any $B \subseteq A$: $\sum_{i=0}^\infty b(i) \le k$ where ...
0
votes
1answer
56 views

Is the set of all sums-of-rationals-that-give-one countable?

Some (but not all) sums of rational numbers gives us 1 as a result. For instance: $$\frac12 + \frac12 = 1$$ $$\frac13 + \frac23 = 1$$ $$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$ Is the set of all ...
0
votes
1answer
47 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
30
votes
7answers
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
0
votes
1answer
61 views

Are there countably many infinities?

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn't there be countably many infinities?
4
votes
2answers
104 views

Infinite shots fired in a lattice forest

A hunter is standing in the center of an infinite 2D forest. There are point trees at all the integer lattice points. The hunter fires a gun with a bullet of zero width in a random direction. He ...
1
vote
2answers
193 views

How big is the size of all infinities?

"Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..." What does exactly mean? How many infinities are there? I've heard there are ...
2
votes
1answer
73 views

What are the factors of $\aleph_0$?

Extend the system of positive natural numbers with $\aleph_0$. Then we have: $$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$ Does it make sense to talk about factors of $\aleph_0$? ...
6
votes
2answers
186 views

Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?
1
vote
1answer
49 views

$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
1
vote
2answers
107 views

Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
4
votes
1answer
109 views

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
6
votes
1answer
282 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely ...
0
votes
1answer
105 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
3
votes
1answer
182 views

“Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? ...
6
votes
2answers
1k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
14
votes
3answers
1k views

The Aleph numbers and infinity in calculus.

I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus? Are they interchangeable/transposable in ...
3
votes
4answers
585 views

What infinity is greater than the continuum? Show with an example

The diagonal argument establishes that the continuum is greater than countable infinity. What is an example of the next infinity (or any greater infinity) and how can it be shown that there is no 1:1 ...
1
vote
3answers
403 views

Comparing infinite sets (of real numbers)

If $A$ is the set of all real numbers in $(0,1)$ with no $5$ in their decimal representation, and $B$ is the set with no $34$ and no $76446$. Then the set $B$ is in some sense larger then $A$, how can ...
5
votes
1answer
598 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
55
votes
6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...