Infinities

[Rule]

What do you think:

Is the set of natural numbers (integers >= 1) smaller than the set of integers?  (Of course, both have an infinite number of elements).

What about the set of integers and the real numbers?

[Sidoh]

This is Cantor's Set Theory, yes?

I did some light research on the subject during school this year and found the little I discovered fascinating.

Since infinity is unbound regardless of the starting vantage point (ie 1 -> infinity or 1,000,000,000 -> infinity), all infinite sets have the same cardinality (size), if I remember correctly.

[dark_drake]

I believe the sets would be the same size. I can remember writing a paper on this, and I discovered this in my research.  It was mind-boggling to think of at first, but it's just one of the problems that arise when humans "with our finite minds try to understand the infinite."

Imagine pairing each integer with one from it's subset, the natural numbers.  For every integer you can think of, there will always be a natural number to pair it with.

[d&q]

http://en.wikipedia.org/wiki/Aleph_number

From what I understood of this article, an infinitely large set is different than infinity as we relate to large numbers. And these infinitely large sets can be 'larger' than other infinitely large sets.

[iago]

The set of all integers and the set of all natural numbers are the same "size".  However, the set of rational numbers is bigger, and the set of irrational numbers is bigger than that.  Or something.

[Rule]

Interesting responses!  You're all right in some ways, and all wrong in others .  But in my opinion, a lot of what makes someone "right" in these questions is whether they conformed to a mathematical definition that I think is debatable (but isn't really debated) -- the definition of "equinumerous."

First of all, the set of natural numbers {1, 2, 3, ..., N} is equinumerous with the set of integers {-N, ..., -2, -1, 0, 1, 2, ..., N}  (as N--> infinity).  This is by definition:  two sets S and T are called equinumerous, and we write S~T, if there exists a bijective function from S onto T. 

I actually don't like this definition.  It is not at all intuitive to me that we should think of the natural numbers and the integers as having "the same size" or "same number of elements."  Imagine generating an integer at random, (say using a computer).  The probability that the integer generated would be a natural number is about 50%, but the probability that it is an integer is 100%.  So, I think of the integers as a bigger infinity than the natural numbers. 

Here would be my definition if I were to recreate mathematics:
Rule's Proposition:  If a set A is a proper subset of a set W, then the cardinal number of A is less than the cardinal number of W.

We probably defined "equinumerous" in the way we did to save ourselves trouble in other areas of mathematics.   So let's use the mathematician's definition.

What does bijective mean?  A bijective function is both injective (one-one) and surjective (onto).  I think the names in ellipses are far better than the latin words, "injective," and "surjective" because you get a good idea of what they mean from what they're called.

So, in most cases (but not always, look up the rigorous definition of function if you'd like..), a function is a mapping from one set to another.  Here are some examples of what it means to be one-one or onto using
functions.

1)
f(x) = x²  will take any real number to a positive real number.
So, f(x): R-->R⁺.  If R is our whole domain, every element of our range R⁺, will be hit, so in this case f is onto (surjective).
But not every real number will have a unique mapping onto the positive reals.  For example,
f(-4) = f(4) = 16.  Or, more generally f(-a) = f(a).  So the function is not one-one (it is not injective).

2)
A function like f(x) = x, f: R --> R  is both one-one and onto, so it is bijective.


So, is the set of even numbers and the set of natural numbers the same size?  YES!  This is surprising, but not hard to prove.  Let's find a bijective function that maps the natural numbers onto the even numbers:

f(x) = 2x .   f: N -->  even N

Loosely speaking the cardinal number of a set is the number of elements it has.  The cardinal number for the natural numbers, or the even numbers, or the integers is all the same! It is aleph_0, which we refer to as the "smallest infinity."  aleph_0 has a lot of interesting properties: aleph_0*r (r>0) = aleph_0,
aleph_0 + any finite number = aleph_0, etc.  But aleph_0^aleph_0 = c -- contiuum, or the next level of infinity!  There is aleph_1, aleph_2, aleph_3, ... aleph_n!! (I sketched a proof that there are an infinite number of infinities of different sizes below).

So, what about the rational numbers?  Are they are a larger set than the natural numbers?  The answer is no -- the cardinal number of the rational numbers is aleph_0.  Try and find a bijection from the natural numbers to the rational numbers.   

The reals are a larger set -- it has cardinal number "c".  I can prove this if anyone's interested, but it would take a page or so.

What about the irrational numbers?  Well, the the real numbers is the union of the set of rational numbers and irrational numbers.  If both the rational numbers and irrational numbers were countable, then their union would also be countable.  But the reals aren't countable, so the irrational numbers aren't countable, and they are a larger set!

The power set is the set of all subsets of a set.
For example,  if S = {0, 1, 2}, then P(S) = { {null}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2} }

Let |A| stand for the cardinality of a set A.

One can show that for any set S, we have |S| < |P(S)|
So,

aleph_0 = |N| < |P(N)| < |P(P(N))| ....
where |P(N)| = c.

Is there a number w, between aleph_0 and c?
Cantor had the conjecture that there is no such set with such a cardinality, and this is called the continuum hypothesis.  It is included as the first of Hilbert's famous 23 unsolved problems (published in 1900), and is still unsolved.
 

There are many books written about infinity.  One that kind of caught my attention was called "unravelling the mystery of aleph_0" .  Sounds magical .


 

[Sidoh]

Cool post! (I read the entire thing!  Usually, a wall of text critically strikes me for 10,000 damage and I die).

Hehe, guess I remembered incorrectly.

I really love this kind of stuff.  Post more of it if you ever feel up to it!

[Nate]

http://en.wikipedia.org/wiki/Aleph_number

From what I understood of this article, an infinitely large set is different than infinity as we relate to large numbers. And these infinitely large sets can be 'larger' than other infinitely large sets.

If I remember correctly, that only applies to sets which are countably infinite.  Basically sets that have a finite beginning and end but an infinite number of data points in them.

[Rule]

http://en.wikipedia.org/wiki/Aleph_number

From what I understood of this article, an infinitely large set is different than infinity as we relate to large numbers. And these infinitely large sets can be 'larger' than other infinitely large sets.

If I remember correctly, that only applies to sets which are countably infinite.  Basically sets that have a finite beginning and end but an infinite number of data points in them.

That's not what a countably infinite set is.

A set W is denumerable if there exists a bijection from the natural numbers to W.  (g: N -->W)
If a set is finite or denumerable, it is countable.

A counterexample to your claim would be the set A = [0, 1] of real numbers.  It has the terminal points 0 and 1, and an infinite number of elements between them, but isn't countable.  There actually is a pretty cool proof for this that I might type up later.  It might be fun writing a series of introductory articles on interesting concepts in mathematics and physics.  This could be a starting point, so ask questions if something seems really unclear   
 

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line, or an extremal point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

I think this is very misleading.  Remember that anyone can contribute to wikipedia.   
Infinity is not "just infinity," in any context (not just set theory).

[d&q]

Edit it then! 

[nslay]

Ooo Ooo
Rule, is there such a set S s.t.
o(N) < o(S) < o(R) ? *wink*

o(N)=aleph_0
o(R)=c

[Rule]

Ooo Ooo
Rule, is there such a set S s.t.
o(N) < o(S) < o(R) ? *wink*

o(N)=aleph_0
o(R)=c

Already addressed this. 

Is there a number w, between aleph_0 and c?
Cantor had the conjecture that there is no such set with such a cardinality, and this is called the continuum hypothesis.  It is included as the first of Hilbert's famous 23 unsolved problems (published in 1900), and is still unsolved.

Trying to provoke a breakthrough in mathematics?