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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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M8dude t1_j29dbmx wrote

well said, although it should be the bijection 1/(x+1) for OP's claim, but that's me nitpicking.

also i think it's natural to assume that OP is talking about the real numbers and the 'counting measure'.

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jaydfox t1_j29ejxx wrote

I think they meant (1/x)-1, not 1/(x-1)

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M8dude t1_j29f874 wrote

aaaah (1/x)-1 is the inverse of 1/(x+1), i mixed up the sets and thought there'd be a mistake, my bad.

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alukyane t1_j29eswo wrote

(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. :)

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M8dude t1_j29ftq5 wrote

yes, you're right, my mistake, but to justify, (1/x)-1 is the inverse of 1/(x+1), so there we are :P

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mrezar t1_j29h13o wrote

Yes but you have to define your domain. For Z its just false, for R it's true.

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ThePhilosofyzr t1_j29ff67 wrote

Not a mathematician here, is this how 2+2 = 5 when the limit of 2 goes to infinity?

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Aki_The_Ghost t1_j29gh9u wrote

No, the limit of x going to infinity of 2 is just 2.

2=2×1=2×X^0 , so as X goes to infinity, X^0 is still 1, so 2 is still 2 and doesn't change.

"the limit of 2 goes to infinity" doesn't make sense, 2 can't go to infinity, 2 is just 2.

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alukyane t1_j29h3oy wrote

Who said anything about x?

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Aki_The_Ghost t1_j29yjfr wrote

We need to know what a limit is. It is the value that some variable approaches but never reaches if one of their parameters approaches a certain point. Example : When X approaches but never quite reaches 0, X^2 approaches but never quite reaches 0. It's the limit of x going to 0 for X^2 . The Limit. A limit implies the constant movement of some variable towards a certain point. So we NEED to talk about X, or a variable of any kind, if we want to calculate a limit. He just said the number 2, a constant number, not a variable, we can't work with it.

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seboll13 t1_j29lr6k wrote

But this would be assuming both sets are countable, right ? But here they are not, so what can we say ?

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alukyane t1_j29m97s wrote

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

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