yeah, there's plenty logic in your argument, there are many different so-called 'measures' to quantify sets of numbers, example the 'distance measure' (i think) of an interval [a, b] is just denoted by b - a.

This is makes the sets have different measure (and more useful ones than just "infinite"), even though they have the same number of numbers.

the measure we would have used before is called the 'counting measure', telling us we'd have to count to infinity for both sets, but that doesn't mean they have the same number of elements (see cantor's second diagonal argument, or yours with the 15), so it has to be shown using a so-called 'bijective function' (our correspondence function), which thank god is pretty easy to construct for any two intervals.

But anyway, good thinking and yep you are right about the example set including [1, 3] and 15, for the counting measure.

it's because the statement in the video maps (0,1) to the natural numbers, not the real numbers.

Veritasium is right with what they said, but since we're talking about 'numbers', generally we're talking about real numbers, not natural ones.

it's a completely different statement.

any element of a set of numbers is a number.

of course, how would you know, never having heard of set theory.

you have to use a different 'correspondence' function for (0,1) and (1,3), which in this case is 2*(x+1/2).

notice how __every__ number in one set has to be matched with __exactly__ one of the other set.

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

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.

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

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'.

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.

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.

yeah that's Cantor's second diagonal argument for more real numbers 0 < x < 1 than natural numbers.

however, there are just as many real numbers between 0 and 1 as there are real numbers at all.

oh i thought you were sarcastic, sorry about that..

Look up the definition of 'Number'.

here, i'll help you:

> https://en.m.wikipedia.org/wiki/Number

and if that's too difficult, here's the relevant part:

> In mathematics, the notion of a number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ...

no.

you're making your own definitions, which is fine, but there are conventional definitions for these things in maths already, and (almost) everyone uses those.

you're thinking of natural numbers.

fractions of natural numbers are still numbers, as anyone who knows any actual maths will confirm.

yet pi is a number.

checkmate, atheists.

yeah clearly the set of rational numbers consists of almost no numbers. /s

lemme try.

for every x in R such that 1 < x < infty, there is exactly one number (1/x) in R, s.t. 0 < (1/x) < 1.

also vice versa.

there's a bijection between the two sets, therefore they are the same size.

am i missing something?

aight you do you then

no clue what you're on about.

if you'd build a tree out of cardboard because you want to grow apples, but instead it grows pictures of apples on paper, THAT would be ironic.