Cantor’s Paradox

Question

I heard about it while watching a TV show, and even after reading through the wikipedia I cannot understand it. [Here](https://en.m.wikipedia.org/wiki/Cantor%27s_paradox) is the wiki for reference. I’d like to understand what the “paradox” behind this is because these sort of things bother me and I feel like I won’t be able to continue without understanding. Thanks in advance!

In: 2

Answer 1

ELI5 will be difficult, as this touches to formal set theory.

In set theory, everything is represented as a set, including numbers.

In this case, a natural number (non-negative integer) is just a set with a specific structure: ∅ (the empty set) is 0, {0} is 1, {0,1} is 2, etc. So each natural number is basically the (finite) set of all natural numbers before it.

A cardinality is an extension of these natural numbers, it answers the question”how many elements are there in this set?”. So N, the set of all natural numbers, is the cardinality of the natural numbers. It’s how many natural numbers there are, quite literally. This is obviously infinite and things get weird quickly, but there are actually more than one infinite cardinality. For example, there are more real numbers than natural numbers.

So each cardinality is basically the (possibly infinite) set of all cardinalities before it.

Now, let’s say we are able to construct the set of all cardinalities.

What is its cardinality? Well, whatever it is, obviously, it is part of the set. And that cardinality would contain all the cardinalities smaller than it.

Since the set contains all cardinalities, its cardinality will need to contain at least all of these as well.

This is probably the most hand-wavy bit: if the cardinality didn’t contain all the cardinalities, then there would be at least one missing, which means either it would be smaller than the set or it would not contain all the cardinalities smaller than itself (and not be a cardinality).

So the cardinality of the set of all cardinalities contains all the cardinalities. Which means it has to contains itself.

This is forbidden by the axioms of set theory (axiom of regularity or foundation). So the set of all cardinalities cannot exist.