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#1 2012-10-20 17:19:43

Calligar
Member
Registered: 2011-09-24
Posts: 234

Not All Infinities Are Equal

I've heard something to the degree of, "not all infinities are equal," mentioned several times and am confused about what this actually means.  The reason why I am asking is because this doesn't make any sense to me at all.  Infinities don't have a numeric value, simply put, they are defined as endless.  How can endless be different then another endless in a weird, simple way of putting it (sorry, stating I am putting it in a weird, simple way, not asking for the answer to be in that way)?

Last edited by Calligar (2012-10-20 17:22:21)


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#2 2012-10-20 17:50:40

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,365

Re: Not All Infinities Are Equal

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


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2012-10-20 18:35:31

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

Okay, I understand what you saying (well, I'm confident I do anyway...), but I don't see how that answers what I'm asking.  Neither N or S are equal to infinity (unless I am mistaken).  You see, how I'm looking at it, they are different, they are rather using infinity to represent how they are endless, or infinite, but they are not infinity themselves.  N represents all the whole numbers, S represents every fifth whole number, though they are endless, neither are infinities, because infinity itself is a different concept (or at least what I thought it was).  For example...

∑ n
n=1
(sorry, wasn't sure how to write it on here...)  This is not an example of what infinity is equal to, but rather a way to use infinity.  Now this is more the reason why I'd argue how neither N or S are not equal to infinity, as this is similar to N, it is rather adding all the numbers infinitely, rather then listing them.  Unless I'm mistaken, you couldn't replace ∞ with N or S at all (it would be very different is what I'm saying).  Now at the beginning you said...

Some sets are denumerable ( countable ) and some sets are not.

Correct me if I'm wrong, but reasons such as these, I disagree with that statement.


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#4 2012-10-20 18:49:05

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,365

Re: Not All Infinities Are Equal

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.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2012-10-20 19:40:33

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

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.

I fear as though you misunderstand where I am getting at.  You see, I knew it was something I either didn't fully understand about infinity or this idea was arguable.  It is exactly why I brought this up in discussion.  I am not going by my own definition, but rather what I already understand infinity is.  If I am actually mistaken about this, I'd like to know why I am wrong about this.  You seem to think I am going by the wrong definition of infinity, where I have spent quite some time understanding it myself, so if you'd be able to explain where I am misunderstanding it, that would be helpful.  So to sum up all of that, what I am arguing is the definition of infinity everyone else is using, and if I am wrong, I like to know why so as I can understand it better.

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

Now I am a little confused by this.  Are you saying that according to set theory, that this is also considered to be equal to infinity?  The reason I'm asking this is because I would otherwise think that this is incorrect, or at least the way you put it.  From what I understand of infinity (the concept I thought was the correct usage), this would be considered a use or example of infinity, but isn't actually infinity itself.  If I am correct about that, then that doesn't answer my original question, where I am trying to make sense of that statement, which I personally feel is not true unless I am misunderstanding something.  However, if that is false, and is according to set theory that that is infinity, then can you better explain to me why that is?  Maybe a proof or something.  Just in case, I'll also reread set theory, to make sure I understand that as I thought I did...

In order to make progress here you are going to have to abandon the concepts you were discussing in earlier threads about infinity.

I am sorry, I am not familiar where this was particularly discussed before.  If you are talking about where I argued the infinity and infinite idea with ssybesma on http://www.mathisfunforum.com/viewtopic.php?id=2079, I am not referring to the same thing at all.  If you are talking about something else, could I please have the link so I know what you are talking about?


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#6 2012-10-20 20:08:16

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,365

Re: Not All Infinities Are Equal

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.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2012-10-20 20:08:22

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Not All Infinities Are Equal

hi Calligar

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.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#8 2012-10-20 20:50:35

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

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.


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#9 2012-10-20 21:14:16

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Not All Infinities Are Equal

hi Calligar

Ok.  I've had a google for a good explanation of Cantor's proof and found loads that were 'explanations', but none that I would promote as 'good', in the sense of clear and understandable.  So my offer to show you the proof still stands.  smile

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#10 2012-10-20 21:21:28

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

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.


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#11 2012-10-20 21:25:56

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

Ok.  I've had a google for a good explanation of Cantor's proof and found loads that were 'explanations', but none that I would promote as 'good', in the sense of clear and understandable.  So my offer to show you the proof still stands.

Sorry, I posted too late.  If you would rather show me and/or explain the proof yourself, if my above statement about it was wrong, I am not against that at all.  Perhaps it would clear up exactly where I am misunderstanding this.  So, in other words, yes, I would like you to to explain to me yourself what the proof is if you feel it would help.


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#12 2012-10-20 21:34:28

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Not All Infinities Are Equal

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.


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#13 2012-10-20 22:10:34

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Not All Infinities Are Equal

So how do you prove that two infinite sets are the same size, or that one is larger?

Cantor's test was to see if you can create a 1:1 correspondence between elements, one taken from each set.  If you can do this so that every member of set A is linked to a single member of set B, and that every member of set B is linked to a single member of set A, then Cantor said the two sets are the 'same size'.  That's his definition and, as I said in the previous post, once you've made a rule, you can test it to see if anything useful results.

Here's a demonstration that {counting numbers} is the same size as {the fractions between 0 and 1}

See picture below.

You can put the fractions into a grid as shown.  If you extend the grid far enough you will eventually get to every fraction.  And you can 'count them' by following the zig zag path shown in red.

The 1:1 correspondence would be

Since the red path goes on for ever, that means that eventually every fraction is paired with a single counting number, so that establishes the 1:1 correspondence.

Now to the main proof.

Let's say you have succeeded in pairing every real with a counting number.  Then you could write a list of the reals and associate the first with counting number 1, the second with counting number2 and so on.  Now it doesn't matter what reals I write where since the argument works whatever order you have.  So the order that follows is arbitary.

1 ----  0.7234561992.......
2 ----  0.234210101023......
3 ----  0.91928376452121......
4 ----  0.450303048576..... 

etc

Now consider the real 0.5482.....  I am constructing this number by making its first decimal place anything that isn't the first decimal place of number 1, its second decimal place anything except the second decimal place of number 2, its third decimal place anything that isn't the third decimal place of number three, its fourth decimal place anything that isn't the fourth decimal place of number 4 and so on.

I chose 5 as it isn't a 7.
I chose 4 as it isn't a 3
I chose 8 as it isn't a 9
I chose 2 as it isn't a 3

The number created by those rules cannot be anywhere in the list, because it has the wrong first digit to be the first number, the wrong second digit to be the second, the wrong third digit to be the third, the wrong fouth digit to be the fourth .........

So there's an real that isn't in the list.  In fact is was dead easy to construct that real; I could easily have constructed many more (an infinite number more !)  so there cannot be a 1:1 correspondence between the counting numbers and the reals.  There'll always be more reals.

Conclusion:  in the Cantor sense, the infinity of reals is larger than the infinity of counting numbers.

Bob

View Image: counting rationals.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#14 2012-10-20 22:15:46

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

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.


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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#15 2012-10-20 22:25:29

bob bundy
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Registered: 2010-06-20
Posts: 6,380

Re: Not All Infinities Are Equal

I don't know how to separate numbers from sets.

I can talk about a set of sheep in a field and, having counted them, I know there are 100 sheep.

Or I can talk about the set of equal pieces of a cake, and, as the cake is divided into 6 equal pieces, and I've got one piece, I've got 1/6 of the cake.

If I try to leave out any consideration of sets, then I'm left with meaningless squiggles.  For me, '3' has no independent existence.  I can count three objects but I cannot have a '3' on its own.

I think that's why bobbym and I keep coming back to sets.  If you can tell me how to consider a number without a set context I'll think again.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#16 2012-10-20 22:31:36

Calligar
Member
Registered: 2011-09-24
Posts: 234

Re: Not All Infinities Are Equal

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 (2012-10-20 22:34:59)


Life isn’t a simple Math: there are always other variables.  -[unknown]

But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle

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