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[DoomYoshi]

I accidentaly figured out two different sized infinities. There are an infinite number of prime numbers. There are also an infinite amount of numbers which are not prime. However, at any given point after 10, the number of prime numbers will be smaller than the number of not prime numbers. Hence, a smaller infinity. Trippy, huh?

[_r.u.s.s.]

nope, exactly the same range infinity

[DoomYoshi]

Yeah, you're right... both sets have a cardinality of aleph-null. It's just so weird though. At 10, there are 5 primes and 5 not primes. At 20 the ratio is 9:11. At 30, the ratio is 11:19. The prime numbers NEVER catch up! How can their infinities be the same size????

[_r.u.s.s.]

it's infinity it doesn't have "size"

[DoomYoshi]

From above...In the context of measuring sizes of sets,
in which "infinity" means a measurement of the size of an infinite set. In this context, such "infinity" concepts do exist but there are more than one of them, since not all infinite sets have the same size. So there does not exist any one single "infinity" concept; instead, there exists a whole collection of things called "infinite cardinal numbers".

The question is about the cardinality (aka size) of those two sets.

[Japo]

According to that, the size (cardinality) of the set of prime numbers would be the same that the set of non primes, and the same that the set of all natural numbers, etcetera:

For instance, the set of even integers has the same size as the set of all integers. In one sense that may seem surprising, since the second set has all the elements of the first set in it plus all the odd integers as well

This concept is different from the other, and pertains to set theory.

If you ask me, the biggest number is the Atlantic Ocean. It's kind of a 5.

[_r.u.s.s.]

Originally Posted by Japofran

According to that, the size (cardinality) of the set of prime numbers would be the same that the set of non primes, and the same that the set of all natural numbers, etcetera

yes that's what we were basically talking about

[AlumiuN]

WTF? How can an infinite set be smaller than another infinite set? They're both infinite! They both have an infinite amount of numbers!

[_r.u.s.s.]

try imagining it on dimensions. 1d line, 2d whole plane and brand new dimension, 3d next noncollinear vector and you've got another dimension with volume

[Japo]

Originally Posted by AlumiuN

WTF? How can an infinite set be smaller than another infinite set? They're both infinite! They both have an infinite amount of numbers!

They're only concepts from set theory, which is based on certain axioms. Axioms are principles that can't be proven, but serve as foundation for the theory. So the theory holds true only after the axioms are accepted as true. And the theory's useful in the extent that those axioms can be applied to real concepts.

To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

The point is, the cardinality of infinite sets is a completely different concept from the concepts of infinity we were discussing before. Let's not mix separate concepts.