Infinity

Polymath257 said:

Hopefully, I'll get to infinities in physics soon.

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metis said:

Please let us know what you learn.

BTW, since the use of math is quite in tune with our universe, infinity does work in some equations.

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Polymath257 said:

So, the first step to a resolution of some of these paradoxes is to realize that we are usually talking about different *senses*
in which something can be finite or infinite. So, the concept of being bounded in one direction does NOT mean that we are bounded
in every direction. So, perhaps to be finite, we require that it be bounded in *every* direction. OK, fine.

Next, that finite line segment whch nonetheless has an infinite number of points. Once again, we are talking about finiteness
in a couple of different senes: one numerical and one geometrical. In the geometrical sense, this line is bounded in all
directions, but the list of points is unbounded.

Similarly, when considering the set of all counting numbers and the set of even counting numbers, we were again looking at two
different ways to compare size: one is containment (where everything in one set is also in the other) and the other is by pairing
off. Now, for *finite* lists of points, these two notions are the same. But, we expect infinite sets to differ in some
ways from finite sets. Perhaps this is one way they do so? That containment and pairing do not need to correspond any longer?
Perhaps the whole is no longer 'larger' than its parts? Depending, that is, on how we measure 'larger': subset or pairing?

So, for convenience at this point, let's just look at sets. We won't think about things like length or area or volume for a bit.
Those can come later. Also, we always consider sets as 'completed'. So the set of counting numbers is an *actual* infinity.

A *proper* subset of a set is just one that is missing some elements of the original. So, the set of even numbers is a proper
subset of the counting numbers. But the set of counting numbers is NOT a proper subset of itself (it is usually considered to
be a subset, though---just not proper).

So, the modern definition of 'infinite' in mathematics is for sets: a set is infinite if it can be paired off with a proper
subset of itself. So, because the counting numbers can be paired off with the proper subset of even counting numbers, the
set of counting numbers is infinite. This definition is attributed to Dirichlet.

So, we can look at some infinite sets and ask if they can be paired off with the set of counting numbers. We already know the
set of even counting numbers can be. We can also pair off the counting numbers other than 1:

1 <--> 2
2 <--> 3
3 <--> 4
4 <--> 5
...
...

How about the set of allpositive and negative integers? Can this be paired off with the regular counting numbers? At first, this
seems impossible, but, intuitively, the counting numbers are 'about half' of the integers and the even counting numbers are
'about half' of the counting numbers, so maybe there's a way. And, in fact, there is:

1 <--> 0
2 <--> 1
3 <--> -1
4 <--> 2
5 <--> -2
6 <--> 3
7 <--> -3
8 <--> 4
9 <--> -4
....
....

In this, each even number on the left gets paired with its half. Each odd number on the left gets paired with the number that is obtained
by first subtracting 1 and then halving, then taking the negative. So, for example, 20 on the left would be paired with 10 and 21 would be paired
with -(21-1) /2 = -10. (corrected)

In this way all counting numbers and all integers are paired off.

How about fractions? Can the collection of all fractions be paired off with the counting numbers? This is much trickier, but it turns
out that it is also possible. I'll show how to do it with just the positive fractions. Throwing in the negatives just amounts to doing
this procedure for even counting numbers and a negative version for the odd ones (as above).

So, we first imagine all fractions a/b where a+b=1. There are not many possibilities here. In fact, if a and b are both at least 1,
a+b is at least 2, so no candidates are seen here.

Next, imagine fractions a/b with a+b=2. There is only one possibility: 1+1 = 2, so a/b=1/1 =1. This is the first fraction to list, so
we pair 1 with this: 1 <--> 1.

Next, consider all a/b with a+b=3: 1+2 =3, 2+1 =3, so we have 1/2 and 2/1. List these in order of the top number, so the second fraction
to list is 1/2 and the third is 2/1 = 2. So, 2 <--> 1/2 and 3 <--> 2.

Next, a+b=4: 1+3 =4, 2+2 =4 , 3+1 =4, so a/b = 1/3, 2/2, 3//1. But, 2/2=1 is already listed, so we only need to add 1/3 and 3/1=3 to our
list.

So far we have

1 <-> 1
2 <-> 1/2
3 <-> 2
4 <-> 1/3
5 <-> 3
..
..

If we continue this, we find that *every* fraction is paired off with some counting number! So, the 'number' of fractions is
the same as the 'number' of counting numbers *if* we are using pairing to identify 'same nt sizes of infinity.

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Polymath257 said:

Now we are in a situation where we have a notion of size even for infinite sets and we actually have two *different* possible sizes
for infinite sets. Are there more?

The answer to this goes through another nice idea in mathematics: that of the power set of a set. Given any set, the power set
is the collection of all subsets of the original set.

For example, if the set is A={1,2,3}, then the following are possible subsets: {1}, {2}, {3} (those with one element), {1,2},
{1,3}, {2,3} (those with two elements), {1,2,3} (three elements)...and one that is easy to miss {} (no elements!). If we count up,
there are 8 subsets of this set of three elements.

Let's do a set with four elements right quick: A={1,2,3,4}. The subsets are: {} (zero elements), {1}, {2}, {3}, {4} (two elements),
{1,2}, {1,3},{1,4}, {2,3}, {2,4}, {3,4} (two elements), {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4} (three elements), {1,2,3,4} (four elements),
for a total of 16 elements of the power set.

Now, we can do this same operation on infinite sets, remembering that many (most?) of the subsets will, themselves, be infinite.
The first neat aspect of this is that the power set of the counting numebrs is the same size as the set of decimal numbers!

But far ore interesitng (in many ways) is the fact that the power set can *never* be paired off with the original set! The power set
is *always* larger than the set you start with! So what? you might ask. Well, we have two sizes of infinite sets already: the
counting numbers and the decimal numbers. And, by what I just said, the second is the same size as the power set of the first.
But we can also take the power set of the set of decimals! This is a set that is *larger* than the set of decimals by our way of
measuring size. But now we can take the power set of that! And the power set of that!

The upshot is that there are an infinite number of different sizes of infinite sets!

Now, a reasonable question is what *size* of infinity is the numebr of sizes of infinite sets? This gets to some very fundamental
problems in the foundations of mathematics. it turns out that you cannot form the *set* of all sizes of infinity! Why not? Because
if you could, you could take the power set of that and have an infinity not in the list! Rather unsatisfying, but without going
into the specific axioms of math, that's where we need to stay.

Since it has been mentioned, we can define things like addition and multiplication of cardinals fairly easily. For example, to
add two caridnals, find sets of those sizes that don't overlap and put them together. The size of the new set is then the sum
of the cardinalities you wanted to add. As an example, we know that the set of even numbers has cardinality aleph-0 (the size of
the counting numbers). SO, and in much the same way, the set of odd numbers also has size aleph-0. Buth putting the two together
gives the set of counting numbers, which has size aleph-0, so aleph-0 + aleph-o = aleph-0.

In fact, it turns out that addition and multiplication of caridnals is easy: the sum and the product of two infinite cardinals
is just the larger one. So, if C denotes the cardinality of the decimal numbers, aleph-0 + C = C and C+C=C.

But, our journey into infinite cardinals and their different sizes leads to a different notion of infinity: one based on order
and not pairing. This leads to ordinals as opposed to cardinals. This will be the next post.

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sayak83 said:

Nice work. Should be a sticky.
All of what you wrote, I could understand quite easily, and I think you can do much more with a better text and mathematics editor. So my question is, where are the good accessible books on transfinite math and other important topics of modern maths (set theory, theory of groups, etc. etc.) that can be understood by folks who know some college math, but haven't gone to pure math. There are some books (by Ian Stewart) that's just fun play with basic maths and then you get to books that start with "Lemma 1, Lemma 2....." too technical with no flow.

Can you recommend something in the middle?

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