The set of all real numbers between 0 and 1 is the same size as the set of all real numbers between 0 and infinity...

>two sets exist
>one of these sets contains the other set, along with a lot of other stuff
>both these sets are somehow the same size though, trust me
Pure math was a mistake. It's all fun and games until it starts flying in the face of actual fucking logic. Thank God we physicists and engineers are there to clean up after the mess the mathematicians leave behind, filter out all the pure autism like "hurr durr muh set theory" and "hurr durr 1 + 2 + 3 + 4 + ... = -1/12 because 1 - 2 + 3 - 4 + ... = 1/4 and muh Reimann", and turn the remaining bits into useful stuff that actually benefits the world.

>both these sets are somehow the same size though, trust me

No one even claims this. They're both uncountable. Nothing to do with size.

>we physicists
You're a fucking retard and clearly not a physicist. Regularization of divergent series is used all the time in physics. See the Casimir effect for instance.
en.wikipedia.org/wiki/Regularization_(physics)
en.wikipedia.org/wiki/Casimir_effect#Regularisation

arctanh is a bijection between the unit interval and [math]\mathbb{R}^+[/math] you fool.

wait wait wait I thought 0-1 was bigger for some crazy mathematical reason
someone on here said something like that

if it sound weird den it make brainne hurt and so it bad

BTFO

>falling this hard for the rival departments meme
Your professors would be ashamed of you

Simpler to make a continuous function; those numbers you're listing are rational.

For every x E [1, inf) there exists a y E [0,1) such that 1/(1-y) = x.

One to one mapping, so yeah, same number of elements.

AN INFINITE CANNOT BE BIGGER THAN ANOTHER INFINITE FUCK OFF WITH THIS STUPIDITY

They're the same cardinality, yeah. Your use of "size" is very cavalier and frankly, unscientific. Get your terminology correct.

The notion of two sets being equinumerous is certainly one of many valid ways of saying two sets contain the same "amount of elements". The idea of a bijection between sets X and Y looks like this: "I can pair every element of X with some element of Y, and vice versa". Not only does this make intuitive sense, it /works/. This method of comparing the "sizes" of two sets does exactly what we need it to. Many properties hold over countable sets and fail on uncountable sets, for example.

There are obviously other ways of comparing the "sizes" of two sets. Maybe you'd prefer to consider the measure (in particular, the Lebesgue measure) of a set. The measure of (0, 1) is 1 and the measure of R is infinity. This notion of examining the "size" of a set also is incredibly useful in multiple fields.

I understand that you want to say that (0, 1) has a lesser cardinality than R, but how much bigger exactly would you like R to be? In your scheme, is (0, 1) smaller than (0, 1]? How much smaller? If you would bother to define some notion of what the "size" of a set should be, we can begin an examination from there. Cardinality does not claim to encapsulate the idea that you want it to, and there's no reason that it needs to.

Intuition starts to break down at infinity. More sophisticated definitions become necessary.

The set of all numbers of the form 1/n where n is a natural number except zero is the same size as the set of all natural numbers without zero.
The former fit into ]0,1] but the latter is unbounded, is this also a problem for you?
Also the traditional notion of size in R^n is the Lebesque measure and they certainly don't have the same Lebesque measure.

It's basically a case of
distance =/= length

here is a demo for kids like you:

hypotenuse length, [math] n \rightarrow \infty [/math]

a+b or [math] \sqrt{a^{2}+b^{2}} [/math] ?

aleph-0
youtu.be/elvOZm0d4H0?t=2m

aleph-1
youtu.be/elvOZm0d4H0?t=4m

...

face it, infinity doesn't equal infinity.

probably cause infinity is NOT ACTUALLY A NUMBER YOU THICK FUCK.

WHATS THE VALUE OF "A LOT" COMPARED TO "MANY" HUH?

SIT
THE
FUCKKKKK
DOWN
STIP LEARNING AND TEACHING RETARD MATH
INFINITY IS NOT A NUMBER
YOU CANT SUM INFINITE SETS IN FINITE TIME
[math]0.\bar{9} \neq 1[/math]
INFINITY HAS NO DISCERNABLE VALUE

>t. retard

>infinity doesn't equal infinity
Dude, even a child can let the fingers on their left hand equal the fingers on their right hand without assigning a number to them.

Pic related. It's prove that things can equal eachother without assigning numbers.

this whole fucking thread