Georg Cantor used set theory to prove that the infinity of real numbers is larger than the infinity of natural numbers.

No, not all infinities are equal in size. It’s a counterintuitive result, but it is logical.

In set theory there are bounded sets {1,2,3,4} which are limited to that range of numbers. Unbounded sets are infinite in range {1,2,3,4…∞}.

Cantor compared sets of infinite series to measure their size.

For example, the infinities of natural numbers and even numbers are equal in size. You can draw a one to one correspondence between each number in each set. So for {1,2,3,4…} and {2,4,6,8…} there is a 1-1 correspondence for 1<–>2; 2<–>4; 3<–>6; 4<–>8… etc. Both sides move towards infinity in an equal 1-1 correspondence. The infinities of Fractions and Natural Numbers are equal in size as well.

Cantor demonstrated that this is not true when he compared a sets of infinite Real Numbers and Natural Numbers. Real Numbers are numbers with an infinite set of decimals including Pi, rational and irrational numbers and so on.

Canto approached the sets and discovered that there is no 1-1 correspondence. There ∞ of reals was larger than the naturals.

You begin the same way by pairing off the real and natural numbers. After doing so, there is real variable that has no corresponding natural pair. He showed why through his Diagonalization Argument, as demonstrated here. In short, after you pair all the natural numbers {1,2,3,4…} with their reals, the reals have leftover series of numbers without a pair. Real infinity is larger than natural infinity.

Here’s the illustration of Cantor’s Diagonal.

There are funny variants of this, like the Hotel Infinity story.

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