That Vsauce video really has done some damage, huh

The smallest infinity is the countable infinity. It is the cardinality (think 'size') of the natural numbers (1,2,3,4,...), hence the name.

Unintuitively, the whole numbers (Natural numbers, 0, and Negatives) have the same cardinality. That means you can match up each natural number with a whole number one-to-one. ('there exists a bijective function')

Even stranger, the rationals (-½,1.3,16.6...) also have the same cardinality as the naturals. The proof is a bit more involved, but still not that hard.

Now, what infinity is larger than others, then? This is where we find the Reals (non-terminating decimals, π, e, √2). No matter what you do, you cannot match them up with the naturals. If you're curious about that, look up Cantor's diagonal argument.

But, interestingly enough, the numbers between 0 and 1 have the same cardinality as the Reals! Any interval within the Reals is the same 'size' of infinity as the entire Reals. You can always find a one-to-one correspondence between the two. (For (0,1) and R you could pick tan, for example)

More generally, if you want to produce a 'larger' cardinality from an existing infinite set, you can look at it's power set. That's the set that contains all possible subsets from the original, and always has a larger cardinality than the old one.

May as well go through the proofs:

First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.

So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.

Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.

Say we have an arbitrary mapping from every integer to a real within [0, 1]:

0 -> 0.89236…
1 -> 0.47389…
2 -> 0.84776…
3 -> 0.18790…
4 -> 0.90542…
⋮           ⋱

This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:

0 ->  …8… ≠ 9
1 ->   …7… ≠ 8
2 ->    …7… ≠ 8
3 ->     …9… ≠ 0
4 ->      …2… ≠ 3
⋮           ⋱

          0.98803… is not within the list

Therefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.

Edit: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

The cardinal numbers have entered the chat. Along with the ordinal numbers, surreal numbers, extended real numbers, projective extended real numbers, wheels, and Riemenn sphere.

That's not true, those two infinities are functionally the same. There are bigger infinities than others, or at least higher orders, but those two are the same order.

there's an infinite amount to real values between 0 and 1 but there's an even greater infinite amount of real values between 0 and 2.

This isn't true. Both of those sets have the same cardinality as the real numbers. Measuring infinities can be weird that way.

They are both strictly larger than the rationals, though.

Yes, but in that case it's the same amount. For every real x in the first interval there is a real y=2x in the second. Also for every real y in the second interval there is a real x=y/2 in the first.

"Infinity" and "number" mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number