Are there more numbers between 1 and 2 than there are between 1 and 3?

No, countable means there exists a injection from the set to the set of the natural numbers. So the question is essentially, if every member of a set is being called in a roll call, will every member have to be present at some time, or do there exist members that won't even appear on that infinite list

false as ℵ1 is not equal to 2^ℵ0, the actual cardinality of the set of numbers between 1 and 2, or between 1 and 3, unless you had prooved the continuum hypotesis.

but you're right when saying the cardinality of the set of numbers between 1 and 2 and the set of numbers between 1 and 3 are the same.

>If 2 sets are uncountable are they automatically the same "size"?
In this case, yes, but not in general. There are "larger" uncountable cardinalities. It's hard to say more than that because it's getting to an unsolved problem - the continuum hypothesis suggests there is no cardinal between the size of N (aleph_0) and the size of R (aleph_1 assuming the continuum hypothesis is true).

Note that I'm not particularly educated in this area so I might have misspoke. But hopefully I've given you some some terms to research.

It's not "unsolved", it's independent of ZFC entirely.

Countable means bijection, not just injection, doesn't it?

{1,2,3} is countable, but you're not likely to find a bijection between {1,2,3} and the set of all natural numbers.

no, there are uncountably infinite numbers between 1 and 2 and between 1 and 3. One isn't really greater than the other because they're both uncountably infinite (assuming we are talking real numbers, not just the rationals).

You see, they have the same cardinality number, which means there exists a bijection between them and that basically means they have the same size.
>If 2 sets are uncountable are they automatically the same "size"?
No. This should help:
en.wikipedia.org/wiki/Cantor's_theorem

Depends on how you measure it. There are uncountably infinite numbers in both sets, so equal in cardinality. Lebesgue measure of each is 1 and 2 respectively.

>There are uncountably infinite numbers in both sets, so equal in cardinality.
No. This is not sound reasoning. They do have the same cardinality, yes, but because they have a bijection between them, not because they're both uncountably infinite. There are other uncountably infinite sets with different cardinality.

but we're talking about two subsets of R, so youre being rather pedantic

yes, fpu ieee floats has twice as many numbers between 0-1 than 2-4

twice? more like [0-1] has as many numbers as between [1, max float]

You didn't say "because they're both uncountably infinite subsets of R", you said "there are uncountably infinite numbers in both sets, so equal in cardinality", which isn't being pedantic to say is wrong. I can list you plenty of sets where your conclusion doesn't follow from your premise.

I would also have let it slide if your explanation was somehow simpler, but I seriously doubt there is a person on earth who understands the distinction between uncountably infinite sets and countably infinite sets, yet doesn't know what a bijection is.

From 1-2 there is an infinite set ranging from 1.000..1... to 2 AND there is also an infinite set which ranges from 1 to 3. You could say that 1-3 would be infinite+1 compared to 1-2 because it includes another infinite set.

The correct answer is neither though.

>"Is X larger than Y?"
>The correct answer is neither

there are the same number of numbers between those two ranges of numbers

f(x) = 2*x - 1 maps [1, 2] to [1,3].
f is continous and strictly monotone. therefore its a bijection and [1,2] and [1,3] have an equal number of elements.

There still exists a bijection between them, even if they're uncountable, so they have the same number of objects.

couldn´t you just take the sets M1 and M2
M1 contains all elements [1,2]
M2 contains all elements [1,3]
Then take intersection from M1 and M2 = M3
then M1 - M3 = empty set and
M2 - M3 = (2,3]
And by that M2 contains more elements (numbers) than M1
I mean all non empty sets contain an infinite amount of numbers, but you can simply take out all numbers from [1,2] from both M1 an M2 and M2 is still has an infinite amount of numbers but M1 doesn´t.

That definition only gives us a meaning for "larger" when one set is a subset of the other. If A = [1,3] and B = [4,5], then neither A-B nor B-A is the empty set.

yeah but in our case [1,2] is a subset of [1,3].
else you would have to find a function that transforms one set into a subset of the other.
my idea is simply that even if you have an infinite amount of elements in two sets and you can take out all element that are equal in both sets, the one that is not empty at the end has more elements.

So which is bigger?
The set of all even numbers or the set of all numbers divisible by 3?

>else you would have to find a function that transforms one set into a subset of the other.
Ok, so let's take A = [1,3] and B = [4,5]. To compare them, we transform A into a subset of B using [math] f(a) = 4+\frac{a-1}{4} [/math], and then observe that B-f(A) = (4.5,5] while f(A)-B = 0. Clearly, then, B is larger than A.

good point

There's an infinite amount of numbers between 1 and 2.
These numbers are also between 1 and 3.
2.5 is a number between 2 and 3 that is not between 1 and 2.
Any number between 1 and 2 you give me belongs between 1 and 3 as well and I give you 2.5 which is one number more than the amount of infinite numbers you give me between 1 and 2.

This question is ontologically incoherent. A "continuum", or "real number line", is a socio-mental construct (i.e. a shared delusion) which has no physical existence. It therefore makes no sense to inquire about how many "numbers" exist between 1 and 2, since to do so presupposes the existence of such an object as a "number". You may as well ask how many angels can dance on the head of a pin.

a philosopher escaped his asylum. someone call campus security.

>You may as well ask how many angels can dance on the head of a pin.
You know that depends on the type of dance user.
743689 for the Salsa.
Only one can b-boy tho, and that shit is fire.