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I don't know, I myself would argue that infinity should not be taken out of mathematics, though I would agree that it can stir up a lot of confusion at times. Infinity is not that hard to understand, the majority of people who misunderstand infinity see it as a number, which is where a lot of the...confusion for many is.
As for the specific use for it, I would argue there it is more likely there is no current necessity for it, but that doesn't mean that that won't matter in the future. The way I see it, everything in mathematics has not yet been discovered, knowledge itself is infinite, we know only a limited amount of it, and that amount we know increases throughout time (this can be said for knowledge in general, not only mathematics). By the way, that last statement is not a fact, it is an opinion, my opinion. I apologize if what I say you don't agree with, that statement was not meant to insult anyone who thinks otherwise.
But anyway, with that, there might be cases where infinity is necessary in the future, on top of that, we do currently use it (though not sure about how necessary it is currently, as bobbym seems to be suggesting it's not necessary anywhere currently). Even with changing the idea to fit better into mathematics, the way I see it, that would in turn be a different concept, still not reason to take out the current infinity concept.
As for the on-topic question, if it weren't cleared up already (which it seems like it has), I can further argue where you go wrong if you are still confused about it.
Hi rvdude,
Welcome to the forum! (or back to the forum...)
From all that I've seen, there are very talented mathematicians here. Even if some of us can't help you, I'm sure there are still a number of those who can, as this is a forum after all. Anyone can answer and if it's not right, someone else is bound to correct it. Hopefully, you will find the help you will need when/if the time comes.
Are you sure that account is deleted and you haven't forgotten your password or something (they give you a password through email at first, then you have to change it yourself). Because checking user list, it says you are still there...
Hmm, I wish I could help, but I'm not too good at ideas like this, anything more math related would be fine. As for shapes, I've always liked higher dimensions (4d +), specifically the pentachoron (for those who don't know what it is, just look at my picture) is my favorite, because it's literally the simplest shape beyond the 3'rd dimension. For symbols, I tend to like infinity, but it isn't as often used, simply (and if I recall correctly, you had that on the site before hand). A puzzle, I can't think of anything off the top of my head that's simple enough to put into a header, maybe mazes (I like them, but I don't personally think it's the best idea for this...)? A game board, um... chess is usually known for being a thinking game. Sorry, just throwing out ideas to possibly help...
I chat about off topic stuff too (or at least I think I do...). I think pretty much everyone does to some degree, but my point is that I do think it would be helpful if it could be done. I was giving ideas because I'm not the original one to bring this up, I've seen others like Shivamcoder3013 bring it up in completely different, from what I saw, mostly unrelated topics too.
I'm not however saying that the forum should be strictly about math, if you look at how the forum is set up, it's not designed to be. But it would also help make some topics better readable. There's a few topics that you go into, and it's conversation is fine at the beginning, but then it tends to go into a different, unrelated discussion, and it can be a little confusing sometimes if they are even talking about the same topic. Later on though, there's like 100 pages of unrelated discussion (an exaggeration for some points). It seems almost futile at that point to even continue to see if there is anything relevant in the topic anymore.
Now off topic conversations are not the only reason I find a chat room to be helpful, that is only one. However, I don't feel it's worth it yet to get too much into that, because it matters if it can be done in the first place or is even a good idea to begin with.
Would he mind if all of us gathered on an un-official chat room on the link I would post somewhere?
Couldn't you do that exact same thing in something like IRC, create a mathisfun channel?
Since everyone seems to go for symbols, why not just use that idea? Maybe make a poll or something on what symbols to use in header?
Hmm, well I can almost assure you math isn't magic, but it is great none the less. As for math challenges, I honestly don't do them much myself because I'm busy studying...newer material, so I continue to learn and understand. Though, once I get far enough, I'll probably start doing more exercises. Though don't get me wrong, exercises can definitely help. They can be good for challenging yourself which can help your understanding of stuff, finding new and better ways of doing things, speed, etc. Of course, if there is something you don't know, that what things like this website is here for.
There is already stuff like that on this website, I'll list a few links..
http://www.mathsisfun.com/games/index.html
http://www.mathsisfun.com/puzzles/index.html
http://www.mathsisfun.com/activity/index.html
All of those things are here on this site, and of course, if he needs any help, he can come to the forum or look around on the website. Like recently, I just played the connect 4 game, and I remember back when I was really young, I used to play connect 4 with my family all the time on weekends; it was one of quite a number of different boardgames we used to play.
Okay, well I guess its not as good of a suggestion then, but is there no way to have a chatroom work at all?
The point though isn't to take that..."freedom" away, but rather finding a better way of doing it. I like this forum too, but I don't see it as perfect. If I didn't like it as much as I did, I wouldn't be on as much. This forum, well, website in general, is thus far my favorite math one I have seen. I in no way would want to ruin anything about it either, but I don't see my suggestion as doing it any harm. My suggestion is merely an idea that I think will help, however, you seem assured that moderating it, even with making it in a members only topic, will still be too much. Are you so sure of this?
I know you were, but I honestly see some potential in it, even if you don't.
Hmm, call me crazy, but I don't think that might be such a bad idea. I mean, adding in the faces of all the administrators could also lead more people into the forum, even recognize them (who have never been to the forum before). I don't know if this is the best way explaining this, but it could help with comfortableness. Though, it is honestly a little bit of a shot too, I'm only saying it might be a good idea, I'm honestly not the most sure. I'm still more thinking something showing math being fun and interesting is better though...
Yeah, I realize it is MIF, it is only a suggestion after all...
People might be able to become members easily, and there might be a lot of members, but how many of these members care to actually get that involved in unrelated discussions. Most of the time, I see most of the same members posting continuous responses, often tending to get into unrelated discussions. I do not see this as too unreasonable; again, this is only a suggestion...
Also, yes, chats do disappear, but a chatroom is just for that, talking, discussing things, etc. You don't need to be on it constantly, because it isn't meant to replace the forum, it is meant for better convenience so members can discuss things with each other if they so choose. It also lowers the flooding of topics.
Hmm, well what if you restricted it? Like say you made a certain forum link where only the chatroom would appear that maybe only registered members can access, similar to the catagory, "Members Only." Perhaps only put it in Members Only or something like that. Because that way, you can be more sure who you are talking to. Chatrooms are in no way perfect, but they are more useful for having discussions rather then flooding topics with unrelated discussions. I know I would personally be more for a chatroom, because there have been quite a few topics I've gone through, they are quite large, but what I notice is that some of them are filled up with unrelated discussions, well, partly filled up anyway.
So far, I think it's a little bit of an improvement. Honestly, I never quite understood the boy/girl thing myself, I went more with the assumption that it was like showing an example of them enjoying math. Anyway, as for the new one, it seems a little more...bland maybe, but I personally feel it's better suiting. As for any suggestions, maybe suggest showing more math like you did on the old one. For instance, you name your site math is fun, so maybe try to put something showing something fun in math, though obviously not being too involved.
As what specifically that should be, I might have an idea... On your forum, you have the category called, "This Is Cool." Maybe have a poll or something to vote for one of these things that would be best for the header? Just an idea, because, that might be a way of showing something interesting and attractive in math, and might be able to show some people how fun and interesting math can be.
What about IRC? I didn't think that was hard to moderate...
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
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.
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.
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.
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.
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?
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.
Well, I doubt your see this considering how old this is, but feel it should be responded to.
You likely hate math because you don't understand it that well. Maybe not seeing any good uses for it, or maybe just thinking it's just another useless thing you have to learn, but maybe consider this... Math itself is pretty much another language, just a more scientifically one, and one derived of from logic. Everything in math is a deductive process to allow us to figure things out, logically, and, at least be more assured that we are correct. Now, just in case, I'll follow up with an example:
Imagine you are sitting next to an apple tree. You sit there, and decide to pick an apple. So you have one apple. Now what are you going to do if you want another apple? You are going to pick another one. This is a very simple example of 1 + 1 = 2. You have 1 apple, and you pick another apple, so how many apples do you have? 2. This is similar to things you do in math (but it gets far more advanced then that).
Without math, we would have barely been able to advance. Many people (in my opinion) don't understand math (or at least not very well...). If you go on simply doing math for the pure reason that you just have to know it, then I might be inclined to agree, not only is math not that fun, but also pointless as well. I do not think I am explaining this the best, but my point is math is so much more than just something you need to know, and typically I find the more one better understands math, the more they tend to like it. You can visualize them as all pointless numbers, or even a stress that makes you have to think, but all in all, it really isn't that bad once you begin to really understand what you are doing.
I for one love math, and only wish to get better and better. There are many things I don't understand yet, but I intend to continue learning, and why? Because there are so many things that I do not know, and it helps me to understand better. If you were serious about wanting to like math more, and to everyone else who agrees with the post above, my recommendation is to start understanding it better. Don't just simply think of them as only numbers you are required to remember, but actually put thought into it, so that you are able to figure out what you are doing. Perhaps, you might find out that, eventually, it helps you in real life (more than you would normally think) too.
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)?