is 51% of infinity the majority of numbers?

In: 13

51% of infinity is still infinity… so technically the answer is yes. That made my brain hurt thinking about that……

Infinity comes in different sizes.

The set of odd numbers is infinite.

The set of all numbers is infinite.

The first set is half the size of the second!

Strictly speaking yes.

On a side note, this kind of situation is very rare in mathematics, as most of the time when comparing infinities one of the two is infinitely larger than the other.

Infinity is weird, because it depends on “how you count”. More precisely, the big property of finite numbers is that no matter how you count them, you will always obtain the same results and not have weird things happening.

It follows that “**51% of infinity**” and “**majority of numbers**” don’t mean a lot.

Let’s start by “**51% of infinity**”. What do you mean by that? You probably meant “every positive integer which is in the first half of the infinity of numbers, so 1, 2, etc up until 51% of the infinity”. Well, the thing is, literally every integer is smaller than 51% of the infinity. There is no integer so that if you double them, you obtain 102% of the infinity, so more than the infinity. So “**51% of infinity**” is literally EVERY NUMBER, so of course it’s the “**majority of numbers**”.

Let’s focus a little more on “**majority of numbers**”. How do we define that? Since infinity is weird, what mathematicians often do is that they make their definition in the finite case. So here you could say “if I stop before infinity at any point, like {1, 2, …, 641} for example, then it’s still a majority of number in this finite slice”. But that’s only one way to do it.

The difficult part when dealing with infinity is that suddenly you have to go into the details of your definitions like a lawyer trying to make a pact with the devil because a small change can lead to very different end results. In a lot of practical case, you have some guidance from the real world (like physics, a lot of mathematical definitions were carefully crafted so that it matches what we observe in practice in physical experimentations). But when you’re talking abstractly about “infinity” and “numbers”, there is not a single universal way of handling the infinity.

Infinity itself is not a number but rather a concept; and thus can’t be divided in any sort of percentage. Half of infinity is ALSO infinity.