They were joking about 2 going to infinity...

Neither of them is countable (in the sense that neither of them can be matched up with the natural numbers).

Who said anything about x?

Fair enough.

Sure, why not!

Fun fact: 2 * 0+2 * 0=5 * 0. Also, 2 * infty+2 * infty=5 * infty.

(1/x)-1 is correct for going from (0,1) to (0,infty).

Your function would send the interval (0,1) to (1/2,1) in a weird reversed/distorted way (check endpoints to confirm).

And the op is most likely talking about cardinalities, not the counting measure, if we're nitpicking. :)

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.

Ok so then what is measurable is local variations in acceleration, not some global acceleration relative to all inertial frames.

And sure in reality uniformly-accelerating frames don't actually exist, but that also includes the zero- acceleration case, since there's always some galaxy far far away applying a force...

We then seem to agree that the top-level claim above about acceleration is wrong: you can't actually tell whether you're in an inertial or accelerating frame, if the acceleration is the same for all observable objects. Right?

It's definitely an accelerating frame, since gravity is acting on it. Probably rotating, too, at least around the planet. In any case I'm mostly interested in how free fall could be distinguished from 0 gravity.

So if they're indistinguishable, I shouldn't be able to measure different "absolute acceleration" in the two, right?

Something's weird in this explanation. Isn't freefall/orbiting indistinguishable from 0-gravity because everything in your environment is experiencing the same forces?