Herkfixer t1_j293yxn wrote on December 30, 2022 at 3:22 PM

Can't be true as stated because, for any "number" between 0 and 1, you will have the same amount between 1 and 2, and 2 and 3, and 3 and 4, and so on. So you would have an infinite more numbers "after" 1 than you have between 0 and 1. Unless you specify whole numbers, integers, rational numbers, etc.

M8dude t1_j295vsc wrote on December 30, 2022 at 3:36 PM

It's counterintuitive, but it's true for real numbers.

the argument is that you can assign a unique real number between 0 and 1 to any real number etween 0 and infinity and vice versa.

Herkfixer t1_j2963q7 wrote on December 30, 2022 at 3:37 PM

Yes, for real numbers. My point is that the OP not specifying the two classes of numbers makes it untrue.

M8dude t1_j29dx0r wrote on December 30, 2022 at 4:30 PM

i think it's fair to assume that OP is talking about the real numbers.

Haven_Stranger t1_j29vkgd wrote on December 30, 2022 at 6:21 PM

Two classes? What two classes? OP specified "numbers between 0 and 1" and "[numbers] from 0 to infinity". The class involved (entailed, even) is "numbers on the number line". In other words, reals. That's one class. That's the basis of the comparison.

Both specified ranges are uncountably infinite.

Here's an easier comparison: There are as many numbers between 0 and 1 as there are between 1 and infinity. It's an easier comparison because now both sets are strictly between their bounds, the two sets don't overlap, and the bijection formula is simpler.

If A is a real between 0 and 1, and B is the matching real greater than 1, then the mapping is:

A -> 1 / B

Also

B -> 1 / A

That's it. For every real number larger than one, there exists exactly one matching number between zero and one, and vice-versa. No exceptions, no excuses, nothing left unaccounted.

The size of the two sets are exactly the same, even though the extents of the two sets are wildly different.

Also also, that's the comparison OP meant to express. Even so, the comparison posted still holds true. It just that, instead of mapping A to the inverse of B, we map A to the inverse of one more than B.

So, no, we don't have to compare reals on one side and rationals on the other, or anything else where we'd have to specify two classes. Those comparisons can be made, of course, but they're not relevant to the post.

It takes different real numbers to make it all the way out to infinity, but it doesn't take more of them. There are exactly as many real numbers greater than 1 as there are between 0 and 1.