Undergrad My basic understanding of set theory

[mr3000]

TL;DR

I’m wondering if this intuition I have is valid regarding set theory

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)…

Is this different from standard cardinality?

 

[FactChecker]

That's a good observation. There are multiple ways of having infinities within infinities within ...
Cantor developed that idea formally. If there is a one-to-one correspondence between two sets, they have the same cardinality. This applies whether the two sets are finite or infinite. There are different sizes of infinity. It can be shown that the rational numbers are the same size (cardinality) as the natural numbers, but that the irrational numbers are a much larger set (see Cantor's diagonal argument).

 

[jedishrfu]

Your observation is encoded in Cantor’s notion of aleph numbers, which measure different sizes of infinity.

The natural numbers have cardinality $\aleph_0$, the cardinality of the countable infinity.

The real numbers have cardinality $2^{\aleph_0}$, known as the continuum.

Whether this cardinal equals $\aleph_1$ is the Continuum Hypothesis, which is independent of standard set theory: it can neither be proved nor disproved.

 

[Hornbein]

Right. Take any two unequal rational numbers x and y. The open set (x,y) -- all rationals between x and y -- is infinite.

 

[Office_Shredder]

A countable union of countable sets is countable.

This means when you say hey I've got infinity infinities, that's gotta be a bigger infinity. Turns out in all your examples here with rational numbers, they are not.