Recent comments in /f/Showerthoughts

M8dude t1_j299cw4 wrote

you're not dumb, it's just counterintuitive.

any number x between 0 and infinity can be paired with a unique number 1/(1+x), which is between 0 and 1.

any number y between 0 and 1 can be paired with a unique number (1/y)-1, which is between 0 and infinity.

there are no exceptions, so none of the two sets of numbers have 'more' numbers in them.

Talking about the 'size' of these sets of numbers is a whole different story.

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alukyane t1_j2998zx wrote

Mathematician here. The op is correct, at least for one common interpretation of "as many".

The usual meaning of "as many" is that you can match up the sets. For example, the interval (0,1) has as many points as the interval (2,3) because I can match x up with x+2.

(0,1) also has as many points as (1,infinity) because I can match x up with 1/x. Or we can match x up with 1/x-1, for the op's claim.

The weird thing is that (0,1) is definitely smaller than (0,infty), in the sense that there are points in (0,infty) that are not in (0,1)... infinity is weird.

The other weird thing is that there are other ways of measuring size that aren't based on cardinality (the pairing up of points). For example, the interval (0,1) has the same cardinality as the interval (5,7), but the two intervals have different total lengths So in that sense (5,7) is bigger... and of course (0,infty) is bigger yet...

So, in "practice" it matters what measure of "more points" makes sense for the particular comparison.

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