A Question on Cardinality $\aleph_{0}$

Question

I'm trying to understand the concept of Cardinality.

My question is,

Let the interval $[1, 2n]$ is given.

In this interval we have $2n$ natural numbers. Or $n\to\infty$, we have countable infinite natural numbers and Cardinality equal to $\aleph_0$.

Then, in this interval we have $n$ even natural numbers. Or $n\to\infty$, we have countable infinite even natural numbers and Cardinality equal to $\aleph_0$.

Then for $n\to\infty$, in this interval $[1,2n]$,we have $$\lim_{n\to\infty} \frac {\text{number of even natural numbers}}{\text{number of all natural numbers}}=\frac 12.$$ In other words number of natural numbers $2$ times many from number of even natural numbers. But, why the cardinalities are equal or What's the point I confused?

Answer 1

Perhaps $\infty=\frac12\cdot\infty$ seems wrong, but $0=\frac12\cdot0$ should illustrate that there are numbers that remain unchanged when divided by $2$. It is possible that $2\cdot\infty=\infty$. What you describe above means that the "density" of the set of even natural numbers is $\frac12$, but the concept of density is different from that of cardinality. https://en.wikipedia.org/wiki/Natural_density One uses limits to define density, but only a bijection (no reference to finite subsets) to define equal cardinality of two infinite sets. The function $f(n)=2n$ provides such a bijection from the set of all natural numbers $\ge1$ to the set of all even natural numbers $\ge2$, so these two sets have the same cardinality.

Answer 2

Envision the famous Hilbert Hotel, where there are an infinite number of rooms numbered $1, 2, \ldots$. All the rooms are occupied.

To create infinitely many more vacancies, the hotel manager reassigns each guest from room $n$ (which they currently occupy) to room $2n$; that is, all the odd-numbered rooms will become vacant and all the even-numbered rooms will still be filled.

Have any more rooms been built? No.

Have any people left the hotel? No.

But there are now an infinitude of vacancies.

Conclusion: The cardinality of the even numbers = cardinality of the odd numbers = cardinality of the set of counting numbers = $\aleph_{0}$.

Answer 3

In Cantor's cardinal arithmetic, $2\cdot\aleph_0=\aleph_0 $.

When dealing with levels of infinity, the rules are essentially the same, but we get some different results (than in the finite case).

Answer 4

First of all, you try to work with limits and want to use that an expression at the limit point equals the limit of said expression as we approach the limit point. But for that you first of all need to know that the function you consider is defined at the limit point. So, how do you define division at infinity? And even if defined, you'd need continuity for your suggested conclusion. E.g., exponentiation $(x,y)\mapsto x^y$ is defined at $(0,0)$, namely $0^0=1$. However, exponentiation is not continous there and therefore we cannot infer $\lim x_n^{y_n}=1$ from $\lim x_n=\lim y_n=0$.

That being said, you should not blindly assume that well-known properties of arithmetic for finite numbers transfer readily to the arithmetic of infinite cardinalities or ordinals.

Final remark: Did you notice that you want to make a claim about $\color{red}{\aleph_0}$ but that limits use the notation $\lim_{x\to{\color{red}\infty}}$ instead?