Are there more real numbers than natural numbers between 0 and 1?

@Cedur216 said in #20:

@tcip reading your post more carefully, I see you are technically wrong with your line of thought, since as you state yourself, "you're not a math wizard".

Your logic (like any other layman's) is: A is a proper subset of B (that is, set B contains all of set A and even more) --> "there are more numbers in B than in A". By that logic, you would also come to give a response like "there are more rational numbers than natural numbers", which is false.

Infinite cardinalities are hard to grasp. We say that infinitely large sets have the same cardinality (or "size", proverbially), if there is a 1:1 correspondence between them: a mapping that sends all elements of A to all elements of B, without missing any element of B, but without using any element of B more than once at the same time. This is also called a "bijection". And with infinite sets, you can sometimes construct bijections even when one set is a proper subset of another.

Take the subset of even naturals as an example (call it E, while all naturals are called N). I will construct a bijection from N to E simply by mapping each n ∈ N to 2n. This obviously covers all of E, but there are no two distinct n, m being mapped to the same even number (as 2n, 2m are also distinct). In a similar way, the integers Z are just as large as the naturals N.
Even the set Z^2 of all pairs of integers is not larger than N itself. If you visualize N^2 as a grid in a coordinate system, then you can draw a "snake curve" that starts in the origin, goes around the angles and doesn't miss any point of the grid. Each n ∈ N will then be mapped to the n-th point of the curve. Since rational numbers are encoded by numerator and denominator (so they're embedded in Z^2), we also see that Q is no larger than N.

But all of this stops working for real numbers, since heuristically, they contain infinite information without any regularity whatsoever. A formal proof goes by Cantor's second diagonalization argument, as laid out in #6.

Thanks for the explanation.

I have to dig in into set theory again. Let me see if I understood the gist of it correctly.

If you can map all elements of a set with all of the elements of its subset, they will always have the same numbers, because mapping of Rn to Zn is possible. Thus there will never be more numbers in either of the sets?

To map a set and a proper subset 1:1, it is necessary that both are infinite. For finite sets, everything is intuitive.

If there's a 1:1 between two infinite sets, then they have the same cardinality. Technically, it means that each set could be embedded into the other one.
Any set that is 1:1 with the natural numbers is called "countable". This is the smallest infinite cardinality. All truly larger sets - those that do not allow a 1:1 mapping with natural numbers - are called "uncountable".

The whole concept of infinity was very controversial throughout the eras of math history, and cardinal / ordinal numbers beyond finite realms are a fascinating field. From our established set theory (ZF / ZFC), we cannot even decide whether there are intermediate cardinalities between the real numbers and the natural numbers.

Well, common sense says there are more integers than even numbers, even though the sets have the same cardinality. I don't think it is good to say that cardinality is length.

@Cedur216 said in #22:

The whole concept of infinity was very controversial throughout the eras of math history, and cardinal / ordinal numbers beyond finite realms are a fascinating field. From our established set theory (ZF / ZFC), we cannot even decide whether there are intermediate cardinalities between the real numbers and the natural numbers.

Actually, it was recently proven that there are no intermediate cardinalities between reals and rationals.

@aentrenus said in #15:

Seems like you (and everyone else in this thread except @JFbin85, frankly) should watch this video, which talk about this exact problem, because your "proof" is unfortunately wrong.

no. my proof is completely fine. but i answered the actual and trivial question that was asked, not a completely different question that makes more sense.

@StephenPS said in #18:

No.of rational numbers = No.of naturals
You can correspond every rational number with a natural.

true. but that is not what op asked. op asked for the count of real versus natural numbers between 0 and 1.

you people can't just make up your own questions, just because op asked something trivial.

edit: well, you can. but those are different questions which have different answers. don't pretend like you asking a different question makes my answer to the original question incorrect though.

@TheCaptain7777 said in #24:

Actually, it was recently proven that there are no intermediate cardinalities between reals and rationals.

You're most definitely not a mathematician and you have no clue. The "continuum hypothesis" is undecidable within ZF/ZFC. You can take both the hypothesis and its opposite as an extra axiom and you have valid systems. That is no "proof".

There are infinite natural numbers, from 0 to ω. But ω is a countable infinity.
First, a natural number can having a friend of integer, such as 1 -> 1, 2 -> -1, 3 -> 2, and so on, 2n-1 -> n, 2n -> -n.
Secondly, a natural number can also correspond to a rational number, such as 1 -> 1/1, 2 -> 2/1, 3 -> 1/2, 4 -> 1/3, 5 -> ...
Also, a natural number can also be paired with an algebraic number(The number that can get by finite amount of arithmetic calculations).
But what if we also include the transcendence?
First, suppose that we have ALL real number between 0 and 1, from a[0] to a[n].
Then, construct a real number b, by this condition(i is the digit index and index of the real number list):
If a[i][i] is not 5, then b[i] is 5, otherwise b[i] is 7.[Reference 1]
Then we got a real number, it is between 0 and 1, but it didn't include in our real number list.
But if we insert b into the list, we can still construct c by the condition above.
Thus, the amount of real numbers from 0 to 1 is infinite and uncountable, much more than countable infinity.
So, real number between 0 and 1 is MORE than the natural numbers.
Q. E. D.
Reference 1: 《Isomorphism: The Mathematics in Programming》, by Liu Xinyu

@JFbin85 said in #14:

going with this that means there are more square numbers than natural numbers. but we can make a 1:1 set up to infinity

1:1 2:4 3:9 4:16 5:25 6:36 7:49 ect.

because we can make a 1:1 ratio they are equal in amount.

I'm not even proving that, what are you even talking about? All I did is prove that empty set is smaller than bigger set. With is good enough proof,

@glbert said in #26:

true. but that is not what op asked. op asked for the count of real versus natural numbers between 0 and 1.

you people can't just make up your own questions, just because op asked something trivial.

edit: well, you can. but those are different questions which have different answers. don't pretend like you asking a different question makes my answer to the original question incorrect though.

Oh wait, I understand what you say now...
I did not read the question properly, sorry.

He does say "sry i meant more real numbers then rational (google translate sucks at this)" later though.