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You are not logged in. #1 20121021 16:19:43
Not All Infinities Are EqualI'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 (20121021 16:22:21) 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 #2 20121021 16:50:40
Re: Not All Infinities Are EqualSome 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. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20121021 17:35:31
Re: Not All Infinities Are EqualOkay, 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...
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 #4 20121021 17:49:05
Re: Not All Infinities Are EqualHi Calligar; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20121021 18:40:33
Re: Not All Infinities Are Equal
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.
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...
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 #6 20121021 19:08:16
Re: Not All Infinities Are EqualThat is the correct link I think. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20121021 19:08:22
Re: Not All Infinities Are Equalhi Calligar You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #8 20121021 19:50:35
Re: Not All Infinities Are EqualOkay, 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).
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.
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... 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 #9 20121021 20:14:16
Re: Not All Infinities Are Equalhi Calligar You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #10 20121021 20:21:28
Re: Not All Infinities Are EqualOne thing I'd like to note, since I didn't make it very clear in my last post...
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...
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. 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 #11 20121021 20:25:56
Re: Not All Infinities Are Equal
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 #12 20121021 20:34:28
Re: Not All Infinities Are EqualCardinality 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'. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #13 20121021 21:10:34
Re: Not All Infinities Are EqualSo how do you prove that two infinite sets are the same size, or that one is larger? 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 You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #14 20121021 21:15:46
Re: Not All Infinities Are EqualHmm, 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...
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)?
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.
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. 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 #15 20121021 21:25:29
Re: Not All Infinities Are EqualI don't know how to separate numbers from sets. You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #16 20121021 21:31:36
Re: Not All Infinities Are EqualSorry, 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). Last edited by Calligar (20121021 21:34:59) 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 