Some sets are denumerable ( countable ) and some sets are not. The notion of countable means to be put into one to one correspondence with N. The set of counting numbers.

For instance the set of every fifth number is the same size as N.

{1,2,3,4,5,6,7...} = N

{5,10,15,20,25,30,35...}=S

They are the same size because there is a function that maps N to the other set. It is 5n. Very, very loosely because there is a function that when acting on N will produce S then S is denumerable and is the same cardinality ( size ).

To better understand what a function is look here:

http://www.mathsisfun.com/sets/function.html

Hi Calligar;

In order to make progress here you are going to have to abandon the concepts you were discussing in earlier threads about infinity. We say the cardinality or size of the two sets is infinity. They are both the same size in the set theory sense. We can not define things the way we want. To discuss this we must speak the same language, have the same definitions. I suggest you adopt even if only briefly the definition that everyone else uses.

That is the correct link I think.

Think of it this way. A long time ago there was a primitive man. He was so primitive he could not speak, he had no numerals at all. But he did have some sheep. He did not know how many he had because he could not count.

Every day he would let his sheep out of their cave and onto the land for them to feed. For each sheep he let out he picked up a stone and put it into a bag he carried. At the end of the day when he would lock his sheep up for the night he would bring them in one at a time. For each one brought in he would take a stone out of the bag. This way he always knew whether he had lost a sheep or not. If the bag was empty, he was happy if not he grabbed his spear and went looking for his sheep.

This is a fundamental concept of how we count sets. By putting then in one to one correspondence with another set. Just like he did it with the stones. You can see this concept even predates numbers.

Same thing with the two sets
1 - > 5
2 -> 10
3 -> 15
4 -> 20
5 -> 25
.
.
.
As long as we can continue to pair them up one number on the left for one number on the right we know that the two sets are the same size. Since the cardinality of N is infinity so is S.

Infinite:definition - Existing beyond or being greater than any arbitrarily large value.

This is as good a definition as possible. N is obviously infinite so is S and they are the same size.

Okay, well from what I've been looking at, it's not that I am misunderstand set theory like perhaps I was thinking.  Though that might still be the case (still have more to check about it to make sure I understood it as well as I was thinking).

As long as we can continue to pair them up one number on the left for one number on the right we know that the two sets are the same size. Since the cardinality of N is infinity so is S.

bobbym, thank you for clearing up what you had meant, I was a little confused about that at first.  Unfortunately, I still fail to see how this relates to my question of how not all infinities are equal.

Rather than thinking of infinities being equal or not equal in size it might make more sense if you think about density on the number line.

There are infinitely many rational numbers along the number line and infinitely many irrationals but it can be shown that the irrationals are more dense.  I'll show you Cantor's proof if you wish but it'll take a bit of typing.

So I think I'm confused on what you mean by density.  Are you saying that because the numbers in S count by larger quantities then in N, that it is a larger, or more dense infinity?  Because the way I'm thinking about this, that only seems to prove that S can be argued that it is larger, or, that might be incorrect.  If I'm understanding this correctly, then that's only saying that S is more dense then N.  Though, I still don't see how this affects infinity...

Sorry, I know this might sound a bit like stubbornness, but I'm really not understanding how these examples in set theory relate to the question.  bob bundy, out of your suggestion, I'm going to look up Cantor's proof to see if that clears up anything.

One thing I'd like to note, since I didn't make it very clear in my last post...

We say the cardinality or size of the two sets is infinity. They are both the same size in the set theory sense.

That was the reason I was thanking you for clearing that up in the first place.  I had originally thought you meant that you were saying N and S were equal to infinity, I misunderstood what you meant until I figured out what you meant by cardinality, which I realized in the other quote...

As long as we can continue to pair them up one number on the left for one number on the right we know that the two sets are the same size. Since the cardinality of N is infinity so is S.

bob bundy, I also looked up Cantor's Proof, from what I understood, it was a proof proving that the infinite sets weren't equal.

Though, I still can't figure out how this affects my above question.  If you are trying to say that infinite sets aren't equal, I don't disagree with that.  I am trying to figure out or argue, why the statement, "not all infinities are equal" make sense.  Unless I'm saying it incorrectly, and saying, "not all infinite sets are equal," then I wouldn't disagree with it.  But an infinite set is another example of infinity, not infinity itself.  The cardinality of an infinite set is equal to infinity, but not the infinite set itself, therefore not proving the above statement correct.

Cardinality of infinite sets is just a way of trying to understand the behaviour of infinity.  As with all maths, you make a set of rules and see if anything useful comes out of that.  Some pure mathematicians don't even require that the results have to be 'useful'.

So Cantor had a way to compare infinite sets.  (I'm not sure if he originated the it but he is certainly the mathematician most commonly associated with the idea.)

Then he showed that some infinite sets are (in that comparison sense) larger than others.  It's an interesting idea; you don't have to like it; but you can accept it's an interesting way of looking at it.

There was a long argument a while back about whether 0.9999999 recurring = 1.  There's a way (that uses the idea of infinity) that shows they are.  Some perfectly respectable mathematicians still don't like that either.

Most people are happy with Euclidean geometry; but, think about it.  When was the last time you drew a line with no thickness, or measured an angle with absolute accuracy, or drew two lines that never meet no matter how far you extend them.  It's just another way of looking at the world.

Bob

ps.  Just read your last.  proof follows.

Hmm, yet, I still don't see the relation of infinite sets to my question, I don't know any better way to put this.  I've tried explaining already that infinite sets don't equal infinity, you even just said...

Cardinality of infinite sets is just a way of trying to understand the behaviour of infinity.

I don't know if there is misunderstanding on my part or the others part, I am simply trying to figure it out, and it almost feels like I am talking about something completely different.  Like from what I'm understanding, it seems like all you and bobbym are doing is talking about infinite sets and trying to prove how infinite sets aren't equal.  Yet infinite sets (unless I am mistaken), are not part of what I am asking about.  On top of that, I don't think anywhere I am misunderstanding what infinite sets are.  Maybe it might help to know what I am getting wrong, if there is anything wrong (like specifically)?

There was a long argument a while back about whether 0.9999999 recurring = 1.  There's a way (that uses the idea of infinity) that shows they are.  Some perfectly respectable mathematicians still don't like that either.

It is also interesting that you mentioned 0.999...=1, because logically I never agreed with this, and have spent...quite a portion of my time arguing it.  Similar to when I first came on here arguing in the post http://www.mathisfunforum.com/viewtopic.php?id=4168.  Later, I argued it with my brother for quite some time, and had come very close to what I thought of as proof, though only to be argued by my brother how it was unprovable, eventually to give up in general with the realization that mathematically, this can't be argued.  Are you saying that this is a similar example, it is another idea of infinity that can not be argued mathematically?  If so, I still need to understand why, it seems completely unclear that this has any proof to it at all.  Yet, with 0.999...=1, I have seen much proof of, though I personally don't agree with it.

Most people are happy with Euclidean geometry; but, think about it.  When was the last time you drew a line with no thickness, or measured an angle with absolute accuracy, or drew two lines that never meet no matter how far you extend them.  It's just another way of looking at the world.

I also do not understand how this relates to anything I've been saying above, in fact, there is nothing what you said I seem to disagree with.  I'm honestly not sure if most people are happy with Euclidean geometry or not.  Drawing a line with no thickness would arguably be 0, which contradicts the idea that it is a line in the first place (in other words, there is no line).  No angle can be measured absolutely in the real world, that is just illogical.  No lines are "truly" straight lines, there is multiple ways to argue this, but it is pointless to proceed.  Nothing of what you said I seem to disagree with at all, and I fail to see the point of that.

Now, to sum this all up simply, both you and bobbym have continuously seemed to talk about infinite sets, but how exactly does this relate to proving all infinities are not equal?  That is the main thing I am not understanding.  The only way I can see any sense in any of this is arguing that an infinite set is infinity, and if that is true, or rather considered true by proof or something, then I can see how that makes sense.  Otherwise, the concept of infinite sets don't seem to relate at all, and if they do in a different way, I fail to see it.

Sorry, posted too late again.  What you have responded (post #13) I believe answers my question, so you can probably ignore my last post (post #14).  So all in all, it seems my perspective of it is a little different.  Though the important part is I can now understand why people think this and that is what I was asking in the first place.  Thank you bobbym and bob bundy, you have answered my question and I understand now what people mean when they say this.  As for any continuing arguments I have against it, there are none; just simply realize that I think about infinity in a different perspective (which ultimately means I can't actually argue this further even if I wanted to).

*edit*  sorry, it seems I yet again posted too slowly

Last edited by Calligar (2012-10-21 21:34:59)