Linked Questions

-2
votes
1answer
99 views

Why is 1 divided by aleph null undefined? [duplicate]

So recently I have been thinking about infinity, and one of the things that I thought of was if you were able to get a defined value for the reciprocal of a transfinite (cardinal) number. So, I ...
0
votes
1answer
61 views

Is there a relationship between the unbounded infinities and uncountable infinities? [duplicate]

When a function f increases without bound we say $f(x)=\infty$. How does this idea relate to, if at all with the infinite sets we study in set theory? To give a better understanding of why I'm ...
0
votes
1answer
45 views

Does mean of uncountable infinite equal to infinite, which is known in calculus? [duplicate]

As I understand, the number of all infinite lentgh sequences which is consist of $\left\{1,2\right\}$ is uncountable. I want to learn, Does mean of uncountable infinite equal to infinite, which is ...
110
votes
8answers
74k views

Is infinity a number? [duplicate]

Is infinity a number? Why or why not? Some commentary: I've found that this is an incredibly simple question to ask — where I grew up, it was a popular argument starter in elementary school &#...
76
votes
6answers
7k 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. ...
18
votes
7answers
3k views

There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
7
votes
3answers
410 views

Is $\lim\limits_{n\rightarrow\infty} n$ comparable to $\aleph_0$?

For example, can we say: $\infty=\lim\limits_{n\rightarrow\infty} n < \aleph_0$? These are two different types of structures. The limit being like the length, extension, or just generic magnitude ...
1
vote
4answers
4k views

The use of ‘infinity’ in a intersection/union of infinitely many sets.

I understand the multiple meanings of infinity, per example, the difference between $\aleph_0$ and the $\infty$ in calculus limits as explained here: The Aleph numbers and infinity in calculus. ...
5
votes
1answer
610 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
3
votes
2answers
765 views

Comparing infinite numbers

Suppose you have 2 infinite numbers, say $A$ and $B$. $A$ is an element of the hyperreals, so that $A$ is greater than every real number. $B$ is the size of the set of natural numbers, $\aleph_0$ ...
5
votes
2answers
409 views

Constructing the reals from fractions of ordinals

We can construct the positive rationals from ratios of positive integers (and thus from pairs of finite ordinals). Can we analogously construct the reals from pairs of countable ordinals?
0
votes
2answers
697 views

Is infinity a real or complex quantity?

Since I was interested in maths, I have a question. Is infinity a real or complex quantity? Or it isn't real or complex?
-4
votes
1answer
1k views

Which one is bigger, infinity sign(∞) or aleph number? [closed]

the infinity sign(∞) is often used casually but it is very abstract concept and ill-defined... when there are 'infinite' natural numbers and aleph-zero is cardinality of a set of natural numbers.. is ∞...
0
votes
1answer
260 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, ...
5
votes
1answer
442 views

Is $2^\infty$ uncountable and is cardinality a continuous function?

I apologize if the title seems too vague, but this is how I was asked the question. So one of my friends intended to write an infinite sum like $\displaystyle \sum_{i=1}^{\infty} a_{2^i}$ . However, ...

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