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I'm a 10th grader looking for a way to understand infinity, I watched a bunch of videos on the topic but I just couldn't get an idea. Can someone explain the idea of infinity or rather how Cantor came up with the idea and his proof for it?

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First off, let me preface this post with the very important note that the notion that Cantor discusses has nothing to do with the term $\infty$ that appears in calculus, nor with the various geometric notions of "at infinity" such as seen in inversive geometry or projective geometry.


Cantor's notion of cardinal number is about quantifying sets up to bijection. We call this quantity the cardinality of a set.

A bijection is a one-to-one correspondence between the elements of the sets: for example, we can write down a bijection between the natural numbers and the primes as follows:

$$ \begin{matrix} 0 & 1 & 2 & 3 & \ldots \\ \updownarrow & \updownarrow & \updownarrow & \updownarrow & \\ 2 & 3 & 5 & 7 & \ldots \end{matrix} $$

Formally, a bijection from a set $X$ to a set $Y$ is a function $f$ from $X$ to $Y$ with the property that $f(x) = f(y)$ implies $x=y$. (here, "function" is meant in the precise set-theoretic sense)

When there exists a bijection between two sets, we say they are equipollent. (or bijective or isomorphic)

In the case of finite sets, it's pretty easy to see that two sets are equipollent if and only if they have the same size — we can just list out the elements side by side. Thus, we can quantify the finite sets by using natural numbers.

Cantor's key insight is that this notion applies equally well to infinite sets: e.g. the set of natural numbers and the set of primes described above are equipollent, as shown by the correspondence indicated above.

Thus, these two sets are quantified by the same cardinal number; we call it $\aleph_0$.

The "surprise" turns out to be that $\aleph_0$ doesn't quantify all infinite sets — there are infinite sets that are not equipollent to the natural numbers! Consequently, there are many different infinite cardinal numbers in addition to the familiar $\aleph_0$.

With the benefit of hindsight this maybe shouldn't be so surprising — it's just that people have spent millenia with just a single word for quantifying non-finite sets, so it's suddenly shocking when it's discovered there are more refined things you can say!

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