Infinity

[Polymath257]

Infinity is a subject that has caused a lot of discussion, insight, and confusion over the years.
My goal in this thread is to go over just a bit of how this concept has been viewed in the past
as well as how modern mathematics views it.

First the definition from Marriam Webster:
-------------------------------------------------------
Definition of infinite
1 : extending indefinitely : endless

infinite space

2 : immeasurably or inconceivably great or extensive : inexhaustible

infinite patience

3 : subject to no limitation or external determination

4 a : extending beyond, lying beyond, or being greater than any preassigned finite value however large

infinite number of positive numbers

b : extending to infinity

infinite plane surface

c : characterized by an infinite number of elements or terms

an infinite set

an infinite series
--------------------------------------------------------

On the other hand, the definition from Dictionary.com gives:
---------------------------------------------------------
1. immeasurably great: an infinite capacity for forgiveness.

2. indefinitely or exceedingly great: infinite sums of money.

3. unlimited or unmeasurable in extent of space, duration of time, etc.: the infinite nature of outer space.

Mathematics.
4. not finite.
(of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

-----------------------------------------------------------

Even here, we begin to see the difficulties of the topic. For example, the immesurably great and exceedingly great categories are actually finite, but just very large. As an example, the number 1,000,000,000,000,000,000,000,000 is very large, but it is not infinite.

The concept of being 'unlimited' is also not exactly to point. For example, the list of counting numbers,

1,2,3,4,5,6,....

is unlimited to the right, but is most definitely limited to the left. The number 1 is the smallest thing in the
list. And, is this an example of 'extending indefinitely' since it goes on and on to the right? Or is it limted
because of what happens on the left?

On the other hand, we have the list of all positive and negative integers:

..., -4, -3, -2, -1 ,0 , 1, 2 ,3 ,4, ....

which is unbounded on both sides. Are they both infinite? Only one?

We can give a host of other examples. So, imagine two perpendicular lines which divide the plane into four pieces. Are the pieces individually infinite? They are each unbounded in some directions, but bounded in others.

There is also an old notion of 'potential infinity' versus 'actual infinity'. The first is suppose to denote a
process that has no ending. So, the list of counting numbers above is 'potentially infinite' because we
would never actually finish the list. If, on the other hand, we want to take the *set* of all counting numbers,
that would be an 'actual infinity' because we consider the set to be complete (we know exactly which things are
in the set and which are not).

Then there are paradoxes where things that seem to be 'finite' under one definition and 'infinite' in another.
For example, take a line segment that is some given, known, length. Say a meter long. This most definitely seems to be finite. It doesn't go on forever. It is noce and bounded. But, there is a half-way point in that line.
Then a half-way point between either end and the half-way point. Then half-way points between all the points so far. And, in fact, between any two points, there is a half-way point that is different than either.

So, even though the line is 'finite' in length, it has an 'infinite' number of points. So, which is it? Finite or
infinite?

And then there are paradoxes just from the infinite sets themselves. Go back to the t of counting numbers. From that extract the *even* counting numbers:

2,4,6,8,10,...

Now, I ask, which has more things? The list of all counting numbers? or the list of even counting numbers? Well, certainly any sane person would say that the collection of all counting numbers is larger. But that leads to a paradox because we can pair off the collection of counting numbers and the collection of even counting numbers as follows:
1 <--> 2
2 <--> 4
3 <--> 6
4 <--> 8
5 <--> 10
...
...

Here, every counting number on the left is associated with its double. Correspondingly, every even counting number on the right is associated with its half. In this way every counting number is paired with exactly one even counting number and vice versa. So, this looks like it means that the number of each side is the same: there are just as many counting numbers and even counting numbers!

It was partly because of these paradoxes (and others) that many people considered the infinite to be inherently paradoxical or even self-contradictory.

Next, the modern view.

 

[Ostronomos]

There is also an old notion of 'potential infinity' versus 'actual infinity'. The first is suppose to denote a
process that has no ending. So, the list of counting numbers above is 'potentially infinite' because we
would never actually finish the list. If, on the other hand, we want to take the *set* of all counting numbers,
that would be an 'actual infinity' because we consider the set to be complete (we know exactly which things are
in the set and which are not).

This idea is essentially correct. By assigning a mathematical value to infinity you have made it complete. For example, a whole of infinity would be one infinity, and thus one has limited infinity to a single set. Hence no paradox. When infinity is added to infinity it yields infinity, aleph_0 + aleph_0 = aleph_0. The question of impossibility arises. But nevertheless we have a limitation of the unlimited set. Only that which is temporally parallel to the infinite can count it.

 

[Altfish]

It is hard to get your head round but makes sense.

It always amazes me that …
Infinity + Infinity = Infinity

 

[Polymath257]

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 number'.

Next: different sizes of infinity.

 

[Polymath257]

This idea is essentially correct. By assigning a mathematical value to infinity you have made it complete. For example, a whole of infinity would be one infinity, and thus one has limited infinity to a single set. Hence no paradox. When infinity is added to infinity it yields infinity, aleph_0 + aleph_0 = aleph_0. The question of impossibility arises. But nevertheless we have a limitation of the unlimited set. Only that which is temporally parallel to the infinite can count it.

It is hard to get your head round but makes sense.

It always amazes me that …
Infinity + Infinity = Infinity

I will get to these. I promise. Some care is required.

 

[shunyadragon]

I believe originally the Greeks developed infinities as a part of philosophy to define the nature of our physical reality.

In math and science today infinities are apart of the tool box like other math and objective verifiable evidence in developing falsifiable theories and hypothesis. Science as such is not overly concerned about infinities even in physics and cosmology except as a part of math. Even though many consider our physical existence 'potentially infinite' science works very well whether it is or is not eternal or infinite.

 

[Ostronomos]

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 infintie. This definition is attributed to Dirichlet.

This would define infinite nestings. Infinity exists without scale or complement, and thus would have no limit. The subset would mean that infinity contains infinity, which is correct. This self-containment gives rise to an analog of self-reference. It is said that since our universe is self-referential, it has the property of self-awareness at varying levels. Another correct deduction.

 

[Polymath257]

At this point, we have a notion of when two sets have the 'same size' as defined by whether they can be paired off with each other
as in the previous pasts. We have found that the 'size' of the set of counting numbers is the same as the set of even counting numbers
and also the same as the set of fractions. This is all using the pairing definition of size.

Now, after a bit, this starts to seem pretty reasonable: we have the finite sets. They can be paired off exactly when they have
the same number of elements, which seems reasonable to call the size of the set. And then we have infinite sets like the set
of counting numbers. And it seems 'reasonable' to expect that all infinite sets are the 'same size', namely, infintie.

But, this turns out not to be the case. There are actually different 'sizes' of infinity as defined by this pairing notion
of size. There are two standard ways to show this. The easiest is my first example. But first, a bit of background.

Most of us are accustomed to decimal numbers, at least to some extent. We understand numbers like 2.7182818 when they come up on
the calculators and can deal with them. We know, for example that 2.718 <2.7182 < 2.719. But, even if we want to deal with
simple fractions, we may need *infinte* decimal expansions. For example,

1/3 = .33333333....
2/7 = .285714285714....
6/9 = .66666....
2/11 = .020202020....

In each case, a simple fraction has an infinite expression if we want to use decimal numbers. On common point of confusion:
1/3 is NOT an infinite number. In fact, 0< 1/3 <1, so 1/3 is very definitely finite (in the sense of being bounded...).
But the *description* of 1/3 as a decimal is infinite. The same is the case for the other fractions above. All of them
are between 0 and 1. But their *descriptions* in terms of decimal expansions are all infinite. Their descriptions as fractions,
though, are all finite.

Now, one property of all fractions is that when they are expressed as decials, the numbers in the decimal expansion eventually
start to repeat. So, 1/7=.142857 142857 142857 ... with the numbers 142857 repeating as a group in that order. A more tricky
example is 1/12 = .0833333..., where it is only the 3 that repeats, not the initial 08.

On the other hand, there are other decimal expressions that do NOT cycle like this. For example, most people know about the number
pi, which is the ratio between the length of the circumference of a cirlce and its diameter:

pi = 3.14159265358979323846264332387950288419716939937510....

it turns out that this expansion never repeats (proving that isn't easy).

Another subtlety. An infinite expansion is always regarded as a limit. We look at finite pieces of the expansion in succession
and see which number they get closer and closer to. So, for 1/3, we have .3, .33, .333, .3333, .33333, .... and as we go farther
and farther out, the decimal number gets closer and closer to 1/3. This is the explanation for a persistant piece of confusion:

1 = .9999999999....

Think about it: look at longer and longer pieces of this expression: .9, .99, .999, .9999, .99999,.... Is there a single number
that these are getting closer and closer to as we go farther and farther out? yes! and that number is 1. This is what a decimal
expansion of a number *means*. There can be more than one way to express the same number in decimals. But this shouldn't be too
disturbing. The same happens with fractions: 2/4 = 1/2 = 5/10.... If anything, the situation is worse for fractions!

So, now, let's consider the set of all decimal numbers between 0 and 1. We ask if this set can be paired off with the natural
numbers. I am going to try to show that no such pairing can be done: no matter what there has to be some decimal left out of
any pairing with the counting numbers! I'm going to give a specific example, but I hope it is clear that the argument works in
general. So, suppose we have pairing:

1 <--> .32072398370727498....
2 <--> .08379222398723857....
3 <--> .23798559639543632....
4 <--> .79857957972987239....
5 <--> .28752927927497237....
...
...
...

What I have to show is that no matter what pairing I am given, there is some decimal number that is left out. No such pairing
can possibly be complete!

How can I do this? Via what is known as the diagonal argument, which is due to Cantor (as is much of the past couple of posts).

Look at the first number on the list. The first number in the expansion of that first number is 3. Pick any number other than
0, 3, and 9 (we excluse 0 and 9 to avoid the 1 = .9999.. problem above). So, let's pick 5

next, go to the second number on the list and look at the second number in its expression. In this case, that number is 8. Pick
something different than 0, 8, and 9. For example, let's pick 2.

Next, go to the third number on the list and the third number in its expansion. We see 7, so we pick something different than
0, 7, and 9. Let's pick 4.

So, we go down the diagonal, choosing at each stage a number different han that diagonal number as well as 0 and 9. Now, create
a NEW decinal from those chosen numbers:

.5246....

This number can't be on our list! Why not? it differs from the first number on our list in the first decimal place of the expansion.
it differs from the second number on our list in the second decimal place. It differs from the third number in our list in the
third decimal place, etc. There is just no place on this list for this number.

What does this mean?

Well, it means that the number of decimal numbers between 0 and 1 is *larger* than the set of counting numbers! There are different
sizes of infinity!!!!

Now, this notion of size is, once again, based on the concept of pairing. There are *other* notions of infinity that we will want
to investigate.

But for now, let's do some definitions. We say that the 'size' of an infinite set is its *cardinality*. We give a name to the
cardinality of the set of counting numbers: it is called aleph-0. We now know that the size of the set of decimals is not aleph-0!
it might be very tempting to call it aleph-1. But we have to be careful. Is it possible that there are infinite sets whose size
is *between* those two? and if so, then calling the set of decimal number aleph-1 would be an issue. We want aleph-1 to be the
*next* larger infinity after that of the couting numbers.

And here is a great known, unknown in mathematics. It has been shown that the current assumptions of mat cannot resolve the question
of which aleph the set of decimal numbers should be given. Cantor guessed that it should, in fact, be aleph-1. But this is not
proved, nor can it be proved without additional assumptions in math! It is known as the Contiuum Hypothesis.

 

[Polymath257]

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.

 

[Ostronomos]

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 dcimal numbers, aleph-0 + C = C.

Correct. Since, as you noted, the cardinals are individually unbounded, and the set of individually unbounded cardinals, is an infinite set of ultimate size.

 

[Polymath257]

So, the next notion of infinity that we will tackle is that of infinite ordinals. Where coardinals count 'how many', ordinals
will count 'at which stage'. For technical reasons, we start with 0 in this. So 0 is the first stage. The next is 1, the next is 2.
We can keep on going and get the counting numbers (with 0 added in)
0, 1, 2, 3, 4, 5, 6,....

but now, imagine putting another stage immediatey after all of these. And, just like what happened before, the name of the stage
becomes the next thing added on (the first stage was 0, but the next we added on 1. The second stage was 0,1 and we added on 2 for
next). We will call this stage ѡ (the greek omega). So, the stage *after* all of these looks like.

0,1,2,3,4,5.....,ѡ

But now we can add new stages:

0,1,2,3,4,5,....,ѡ, ѡ+1, ѡ+2, ѡ+3, .....


Now, what should the next stage be called? Well, one easy answer is ѡ+ѡ, and this is valid. But another way of looking at it
is that we took one copy of ѡ and doubled it. This gives ѡ*2. But already there is a LOT of care that needs to be taken when adding
or multiplying ordinals. They do not work like ordinary numbers! Why not?

Well, suppose that I ask for 1+ѡ. That would mean taking a first stage thing and putting at the end an omega stage thing. This looks
like
0,0,1,2,3,4,.....
and this looks like an ѡ picture! So, 1+ѡ=ѡ, but these are different than ѡ+1. The order of addition is important here! ,

Simialrly, to multiply, we take a copy of the first for each stage of the second. So, 2*ѡ would look like

0,0,1,1,2,2,3,3,4,4,.....

and this has a stage desription that is ѡ. So, 2*ѡ=ѡ, but ѡ*2 is very different. Multiplication depends on the order also!

Anyway, we get, after ѡ*2 = ѡ+ѡ,

0,1,2,3,4,5,....,ѡ, ѡ+1, ѡ+2, ѡ+3, ...., ѡ*2, ѡ*2 +1, ѡ*2 +2, ѡ*2 +3, ...., ѡ*3

And we can keep on going. After ѡ*3, we eventually get to ѡ*4, then ѡ*5, etc....keep going to get to ѡ*ѡ=ѡ^2.

Then, yes, we can keep going ѡ^2 +ѡ, ѡ^2 + ѡ*4 +1, ..., (ѡ^2)*2, (ѡ^2)*3, ..., ѡ^3, ѡ^4, ѡ^5, ...., ѡ^ѡ.

This keeps on going.

But the remarkable thing is that ALL of the ordinals we have considered can be paired off with the counting numbers! They are all
the same cardinal! They are just different *ordinals*.

So the question arises as to whether there is an ordinal who cardinality is the same as the set of decimal numbers. To answer this
question affirmatively requires an axiom called the Axiom of Choice. Perhaps I will get into that axiom another time, but the basic
content for us is that every cardinality has an ordinal of that size.

 

[Polymath257]

Correct. Since, as you noted, the cardinals are individually unbounded, and the set of individually unbounded cardinals, is an infinite set of ultimate size.

There is no 'set' of cardinals, it turns out. The collection is a proper class.

 

[Ostronomos]

Consider me fascinated.

 

[ChristineM]

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 number'.

Next: different sizes of infinity.

Absolutely fascinating, i can't imagine infinity (yes I've tried) yet alone count it

So i can't see the set of counting numbers is infinite. To make the set of counting numbers the number must be able to be counted, if it can be counted it cant be infinite.

 

[Polymath257]

Absolutely fascinating, i can't imagine infinity (yes I've tried) yet alone count it

So i can't see the set of counting numbers is infinite. To make the set of counting numbers the number must be able to be counted, if it can be counted it cant be infinite.

OK, so the question is 'infinite in what sense'. A finite set is defined to be one where the number of elements is some counting number. So, the the {1,2,3,4,5} has 5 elements, and 5 is a counting number, so that set is finite. The set {3,28, 76, 10298, 477253} has 4 elements in it and 4 is a counting number, so it is a finite set.

But, the set {1,2,3,4,5,...} does not have a number of elements that is a counting number. Hence it is not a finite set.

Your assumption that the number of elements in the set has to be counted to qualify something as a set is the mistake. All (at this level, anyhow) that is required is that we can know when any individual is in the collection or not.

So, for example, I can know that 74729347298 is a counting number without having a list of all the counting numbers. I can know that 2.3 is NOT a counting number also.

One of the assumptions in modern math is that there *is* a set of all counting numbers. To be counted and to be a set are different things. A set is just a collection of other things. If you want to go the axiomatic route, we assume there is an empty set, that whenever you have two things, you can form a set with those two things as elements, that when you have a bunch of sets, you can form their union, that you can form the power set of any set, and that there is an infinite set (I left off a couple of the axioms, but you get the idea).

 

[Polymath257]

Absolutely fascinating, i can't imagine infinity (yes I've tried) yet alone count it

So i can't see the set of counting numbers is infinite. To make the set of counting numbers the number must be able to be counted, if it can be counted it cant be infinite.

http://googology.wikia.com/wiki/Googology_Wiki


I actually have more difficulty 'imagining' very large numbers than I do infinity. Yes, I know, strange. But there are techniques for defining *really* big numbers out there...all of which are 'finite'...

Here's a fun website for *large* numbers:

Googology Wiki

 

[ChristineM]

OK, so the question is 'infinite in what sense'. A finite set is defined to be one where the number of elements is some counting number. So, the the {1,2,3,4,5} has 5 elements, and 5 is a counting number, so that set is finite. The set {3,28, 76, 10298, 477253} has 4 elements in it and 4 is a counting number, so it is a finite set.

But, the set {1,2,3,4,5,...} does not have a number of elements that is a counting number. Hence it is a finite set.

Your assumption that the number of elements in the set has to be counted to qualify something as a set is the mistake. All (at this level, anyhow) that is required is that we can know when any individual is in the collection or not.

So, for example, I can know that 74729347298 is a counting number without having a list of all the counting numbers. I can know that 2.3 is NOT a counting number also.

One of the assumptions in modern math is that there *is* a set of all counting numbers. To be counted and to be a set are different things. A set is just a collection of other things. If you want to go the axiomatic route, we assume there is an empty set, that whenever you have two things, you can form a set with those two things as elements, that when you have a bunch of sets, you can form their union, that you can form the power set of any set, and that there is an infinite set (I left off a couple of the axioms, but you get the idea).

Maybe it's the word "counting" that's throwing me

 

[metis]

According to researcher Leonard Susskind, most cosmologists now believe that infinity is most likely and that we are likely a part of a multiverse and not just a universe..

 

[Polymath257]

Maybe it's the word "counting" that's throwing me

Yes, the set is the set of 'counting numbers'....1,2,3,4,5.... But the entire set is not a counting number. Nor does it have a counting number of elements. Each individual is counted, but not the whole set, if you'd like.

 

[Polymath257]

According to researcher Leonard Susskind, most cosmologists now believe that infinity is most likely and that we are likely a part of a multiverse and not just a universe..

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