r/askmath • u/Loose-Balance3225 • May 06 '26

Discrete Math Can infinity contain infinity

If pi has no end it has to have every combination of numbers but could it hold an infinite combination? Like 1 2 3 4... To infinity

Please help this is missing my brain up

• Upvotes

66% Upvoted

60 comments sorted by

View all comments

•

u/idksomerandomcrap May 06 '26

Yes, you don't need complicated math to prove it. You have a set of numbers, 1 to infinity. This set contains all numbers. You have another set that is all even numbers. The set of all even numbers would still be infinite. The set of even numbers will be contained within the first set. There are different levels of infinity.

•

u/shwilliams4 May 06 '26

The set of even numbers is not larger than the set of numbers so it is not contained. They are the same cardinality.

A correct example would be the reals and the integers.

•

u/Bills_afterMATH May 06 '26

Whether it’s “larger” depends on the notion. They have the same cardinality, but the set of even numbers has density 1/2 compared to the set of natural numbers which has density 1.

•

u/tottasanorotta May 06 '26

Cardinality without this idea of density seems very misleading for describing sizes of infinite sets. It seems extremely unintuitive when someone says to me that the even natural numbers have the same size as the natural numbers. You can keep pairing them up to as far as you like successfully, but you can't tell me that there isn't half as many evens for every finite n.

•

u/idksomerandomcrap May 07 '26

Youre so close. Thats kind of exactly the point. Yes there are half as many, but it is still an infinite set. Even though there is half the numbers in the set, half of infinity is infinity. Thus there are different sizes of infinity. You call it density, sure. I put it in laymans terms because reddit users are not rocket scientists.

•

u/Bills_afterMATH May 07 '26

It captures this intuition of being “half as big” for some nice situations like the set of even numbers. Just like measure does. For example, the Lebesgue measure of (0,1/2) is 1/2 and of (0,1) is 1. Just need to be really careful since density is often not well behaved. Probably one of the reasons that it’s not usually introduced in something like an intro to proof course.