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Ecoris
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posted January 13, 2010 12:39 PM |
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Edited by Ecoris at 12:40, 13 Jan 2010.
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I know that it is a matter of definition Ecoris, but it is a rather "counter-intuitive" definition and therefore not ok.
I don't think it makes sense to consider whether the definition is intuitive or not. I have N and N_0 at my disposal. Given the context one will be more convenient than the other. But as definitions they are equally good.
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However, if I have a product like the sum above, I do initialize to 1; e.g. replace in my example above the \sum with \product. That's the intuition of initialization. You have to do it if you want to be consistent.
Why?
If I see an empty product I know that it is 1 by definition. Ask a layman what that empty product is. I bet they would call it meaningless, they do not initialize to 1.
Regarding the lack of a neutral element for addition in {1,2,3,...}:
So? The proper set in which to consider addition and multiplication would be Z not N.
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Anyway, it is a matter of definition, but you don't give me any argument for not including it.
Ok, I have one. But I don't think you'll put as much weight to it as you really should since you seem to care too much for axioms. My argument is not even a mathematical one, it is this: That 0 is not a natural number is a convention in the mathematical society I belong to.
And looking at it from a purely mathematical viewpoint it is more or less just as convenient.
I do not know what you mean by convincing and as said, I don't see how intuition is relevant when deciding whether 0 is a natural number or not.
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Who cares about axioms ?
My impression is that you care too much.
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By the way guys, have you never encountered the other symbolism of N^* ? What is that supposed to mean ?
No. And, I don't know. Z^* = {-1,1} in my world.
Sorry about chopping up your post. It comes to look rather confrontive, but I need to get a better impressions on your views.
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dimis
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posted January 13, 2010 01:48 PM |
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ihor what do you mean you gave me as many arguments as I did ? All I read about your argument, is that (1) it is a definition, (2) natural numbers are there for counting and therefore 0 should not be there. I said I am ok with the definition but I don't agree. As of the second argument, in a while.
Ecoris, I agree, that you can say that the empty product is 1. But we ended up here by "stretching" the notion of counting, right ?
So I ask you guys, the very natural question:
How many posts did you manage to make between this one and ...
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dimis
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posted January 13, 2010 01:48 PM |
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... this one ?
As a note, I also had the impression until about 7 years ago that 0 was not there on N, so I don't remember which definition we had in high school and I can not check right now. I don't think that changing your mind on a definition is that bad. It is just a definition.
Anyway, is there someone who does not know about the problem at hand (0,1) =_c (0,1] and wants to make an attempt ?
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AlexSpl
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posted January 13, 2010 02:29 PM |
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Anyway, is there someone who does not know about the problem at hand (0,1) =_c (0,1] and wants to make an attempt ?
Has this anything with periodic fractions like 0,(9) and geometrical progressions?
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dimis
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posted January 13, 2010 02:38 PM |
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You are very close Alex. I think a geometric progression approach is the best. You somehow want to push f(1) inside your line in (0,1).
Irrelevant: May be alcibiades might be interested in. Apparently, there is even an ISO standard for all these here, sold for a crazy price.
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Corribus
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posted January 13, 2010 06:56 PM |
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I've always refered to natural numbers as the set of integers from 1 to infinity, because that's the way I was taught in, oh, third grade. The inclusion of 0 has, in my experience, always changed the name of the set to the "whole numbers".
But what difference does it make? It's just semantics. Define a set as you please and give it any name you want. The important thing is that everyone discussing a certain problem is using the same set with the same label.
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dimis
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posted January 14, 2010 09:59 PM |
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Edited by dimis at 22:19, 14 Jan 2010.
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Well, yes and no Corribus at the same time.
You can not seriously believe in everything your teachers taught you in primary school. They were thrown in a classroom and they were supposed to teach you language, grammar, literature, maths, physics, chemistry, and whatever else we had. They can not have a well founded opinion about all the above matters. It is understandable and no one blames them about anything.
It is true that it is semantics and it is also true that we want to agree on using the same set with the same label.
However, especially this set which is called "natural numbers" was called like that because it wants to capture - primarily - the very old inherent human need of counting objects in the natural world. And for some of the questions that we form with the language the answer is 0 (I already gave an example above). It is not that we refer with questions to some imaginary objects. The reason is that we refer to "natural objects" which simply happen not to satisfy the context of the question, so they amount to 0.
But the story doesn't even end here. If you go all the way down on the theorem about existence and uniqueness (up to isomorphism) of the natural numbers, you have a cute proof which is based on the empty set ø. Of course, you can mimic the proof and start everything from {ø} -which will not be so cute! - but then, you sort of question your agreement with Zermelo's axioms. It really is Occam's razor. Because if you want to preserve the cuteness of the proof you will start from the empty set ø, which well ... has cardinality 0. You really want to talk about these things later on in Maths. It is as if we play hide-and-seek.
So, if you think I will ever adopt the definition that you guys give (and which I have abandoned by choice in the past), well ... think again. 
Besides, apart from the argument that all the books that give foundations on the mathematics actually include 0 in their definition, I have one of the funniest arguments of all time!  My saying comes with an ISO standard; this one!!! Yay!
Von Neumann once said "In mathematics you don't understand things. You just get used to them." Is this one of them ?
So, do we agree in general, or simply agree to disagree ?
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Corribus
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posted January 14, 2010 10:23 PM |
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dimis, of course I do not accept blindly whatever I learned in grade school. But labels are merely adopted by convention, and I only described the convention that I learned. I mean only this: if you want to refer to the set of positive integers, thus excluding zero, you can call it natural numbers if you want. You can make up a name if it pleases you. Call it the Plotopookian Numbers. It doesn't really matter, so long as everyone else understands what you mean by Plotopookian Numbers. This isn't a mathematical argument - it's a philosophical one. However, if you're going to refer to "Natural Numbers" as a set, and have some sort of discussion about them with someone else, you had better make sure that the other person uses the same convention as you do, or you'll end up discussing different things.
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dimis
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posted January 14, 2010 10:39 PM |
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I agree that we want to give the same label on the same set - always.
However, "natural numbers" is not just a set. It is the very first set people wanted to talk about. It is probably the most primitive one with interesting members and qualities. So, my feeling is that it deserves a little bit more attention when we actually agree on what it is. Simple as that.
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Ecoris
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posted January 15, 2010 12:38 AM |
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"ISO 31-11 is the part of international standard ISO 31 that defines mathematical signs and symbols for use in physical sciences and technology."
Mathematics is not a physical science. Nor is it technology. (Needless to say?) I have never heard of that standard. And why would I need it?
Mathematicians who deal with set theory may see {0,1,2,...} as being more special {1,2,3,...} and therefore reserve the special name "natural numbers" for it. Others may for convenience or whatever use the name for the latter set.
I hardly care at all. It's just names. Like "colour" is spelled "color" in US. It's a matter of culture. I could just as well be used to N = {0,1,2,...} but I happened to grow up in a place where 0 is not a natural number. I don't have any preference based on philosophical arguments and I would be very surprised if your choice of definition, dimis, is not that of your collegues/fellow students.
And what is Occam's Razor doing in mathematics?
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Binabik
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posted January 15, 2010 01:58 AM |
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Lol @ you guys
It's just a freaking word and a convention. Obviously there is no universal acceptance of the definition. Therefore, out of necessity, it must be defined whenever used, especially in an international community. To do otherwise is just asking for trouble.
And for the record, I've never heard of "natural" numbers before. Or maybe it's more accurate to say I've heard the term, but never heard it defined, nor used in context.
On the other hand, it's been 45 years since I learned that stuff, and 30 years since I've taken any math at all. So I could easily just have forgotten it.
And while I'm bragging about loss of memory
I've never heard of "cardinality".
I have no idea at all what "(0,1) =_c [0,1]" is. Although the first thing that pops in my head about the difference between the curvy bracket and the square bracket is inclusive and exclusive, which doesn't work in the context given.
And when it comes to some of the other symbols being used like _c, Z+, Z_+, R_+, R_++, I don't think it's a matter of not remembering. I'm pretty sure I've never even seen those until now. And I'm fairly certain I've never seen an underscore (_) used in ANY mathematical nomenclature.
So could someone explain to me (briefly) what all that stuff is?
Is it maybe computer nomenclature since it's difficult to type characters on a computer other than ASCII (not extended ASCII)?
And speaking of ASCII, I think using a carrot ^ for exponents is really difficult to read. And using a / for division and writing equations in one long continuous line is damn near impossible to read. They don't actually teach it that way in school now do they?
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dimis
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posted January 15, 2010 03:29 AM |
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Edited by dimis at 06:29, 15 Jan 2010.
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Ecoris, Occam's razor is there in my statement because I like simplicity and beauty, and by including 0 things are simpler (and I think more beautiful in the proof; among others).
As of the ISO standard, c'm on!! I would never have expected such an argument in the discussion, I really like it, and therefore:
I S O , I S O , I S O , I S O , ISO, ISo, Iso, iso, is., i.., ...., .. , . , , *runs away shouting 'iso'*
Binabik, you guessed right. In the intervals when you see parentheses then this means that you do *not* include the endpoints. If you see square brackets, you *do* include the endpoints. (I hope this is universal too ) So, it fits in the context, because that equality sign does not really want to say that these two sets are equal, because clearly they can not be; the (0,1] contains 1, while the (0,1) does not ==> Therefore not equal. In fact the (0,1) is a subset of (0,1] (i.e. every member of (0,1) is also member of (0,1]). So, we arrive at that strange symbol =_c. This means write down the equality sign but add to that a subscript of 'c' (due to the word cardinality). So, it says that these two sets have the same cardinality.
An easy proof (depending on what you assume known) is the following:
(0,1) is subset of (0,1] which is subset of (0,2)
hence (by the set inclusions above) regarding their cardinalities you have
(*) cardinality of (0,1) <= cardinality of (0,1] <= cardinality of (0,2) -- Read '<=' as 'less equal' --
On the other hand
cardinality of (0,1) = cardinality of (0,2)
since you can prove this e.g. with the function
g(x) = 2x
i.e. you created a "matching" of the members of the two sets. So, all the inequalities in (*) are indeed equalities and that's the end of the story.
However, the idea of the exercise is to construct such a mapping like the 'g(x)' above directly between the (0,1) and (0,1]; you can start from either one of them. That's the tricky part. My hint was that initially you think of the function f(x) = x which has domain (i.e. x takes values there) on (0,1], but the problem that arises is that f(1) = 1 which clearly does *not* belong to (0,1) and hence you have to define f(1) differently. That's why I was saying try to compress the line by pushing this value point of the graph somewhere else. 
Regarding the rest, I think that the way things are written above you should consider Z_+ the same as Z+. Like before, you write down on a piece of paper 'Z' with subscript '+'. The idea of the underscore is that in order to achieve subscripting when you typically write maths is that this is the command that generates the subscript. The idea of the plus as a subscript means "non-negative", which is kind of funny, because this also means include 0, which is neither positive, nor negative (or you can think of any sign assigned to it; let's not argue about this one now). Similar is the explanation for R+ and R_+. They represent the same thing. R is there for 'Reals' or if you prefer 'Real numbers'. The 'Z' above is there from the german 'Zahlen' and denotes the set of integers.
Oh yeah, the X^* means superscript X with a * and implies that you should *not* include 0. And Ecoris is right; the symbolism is kind of unfortunate, but mathematicians sometimes are not very creative on definitions for the various exceptions ...
As a final comment, when the + is superscript like in Z^+, R^+, etc. then it is as Ecoris says; 0 is not there.
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mvassilev
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posted January 15, 2010 07:44 AM |
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Ecoris
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posted January 15, 2010 06:38 PM |
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As a final comment, when the + is superscript like in Z^+, R^+, etc. then it is as Ecoris says; 0 is not there.
Have I?
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dimis
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posted January 15, 2010 08:25 PM |
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In the previous page you comment on Z_+ and R_+ and I thought that there was a typo there; i.e. Z_+ instead of Z^+. I guess I misunderstood.
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TheDeath
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posted January 15, 2010 08:27 PM |
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Ecoris, Occam's razor is there in my statement because I like simplicity and beauty
Me too, but I can't change conventions 
For the record I didn't even know that cardinality made sense in an infinite set. What is it exactly useful for? Because I find the idea that (0,1) and (0,2) having the same cardinality against basic logic. Like i said before in a post, can you even count to infinity?
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Oh yeah, the X^* means superscript X with a * and implies that you should *not* include 0. And Ecoris is right; the symbolism is kind of unfortunate, but mathematicians sometimes are not very creative on definitions for the various exceptions ...
IMO 90% of all math symbols are unintuitive -- you don't "get the picture" without thinking it through, unlike let's say, some program code
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Ecoris
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posted January 15, 2010 09:19 PM |
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Edited by Ecoris at 21:20, 15 Jan 2010.
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For finite sets, cardinality just measures the number of elements. That's not particularly interesting.
But cardinality allows you to compare sizes of infinite sets: Two sets have the same size (cardinality) if there is a bijection between them. For finite sets, this just means that they have the same number of elements, so you can view it as an extension of that intuitive notion.
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Like i said before in a post, can you even count to infinity?
No. Infinity is just a concept, not a number.
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TheDeath
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posted January 15, 2010 09:29 PM |
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Edited by TheDeath at 21:31, 15 Jan 2010.
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But if you are allowed to construct any function, such as g(x)=2*x, for the bijection to work, doesn't that mean any infinite set has the same cardinality as any other infinite set? (of course I'm assuming infinite sets with finite boundaries)
In this case it is pretty obvious logically that (0,1) and (0,2) are different in size, the latter being greater. If you quantize them infinitely, let's say with a limit, the result of the ratio would be 2, not 1.
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Ecoris
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posted January 15, 2010 09:42 PM |
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To address your questions:
No. Not all infinite sets have the same cardinality. For instance the integers Z = {...,-2,-1,0,1,2,...} has a lower cardinality than the reals R. There is a quite short proof of this.
(0,1) and (0,2) have the same cardinality. One probably pictures the sets as line segments of different lengths, and therefore the latter seems twice as large as the former. However, when you picture them like that you are implicitly including more structure on the sets. Namely their ordering. When dealing with cardinality you only care about them as sets without any structure. That is to say you forget any relations the elements of the sets may have.
There are other concepts of sizes of sets. E.g. there is a notion of "measure" on (certain) subsets of the real line. (0,1) has measure 1, while (0,2) has measure 2. Measure is used in the definition of integrals and it plays a major role in probability theory.
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dimis
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posted January 15, 2010 09:42 PM |
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But if you are allowed to construct any function, such as g(x)=2*x, for the bijection to work, doesn't that mean any infinite set has the same cardinality as any other infinite set? (of course I'm assuming infinite sets with finite boundaries)
No. For example the natural numbers the "size" of the integers is strictly less than the size of the real numbers. It is even less than the size of (0,1). We can come back to this one.
The idea of the bijection is that you take each guy from one set and each guy from the other set, they form a "line" (queue) by facing each other, and every guy on every line is facing a unique guy on the other line; hence you should "count" the same amount of guys on every line! So, it is logical, since as Ecoris said (and Zamfir implied) you can not count up to infinity. There is no such number.
Alright, I will leave the problem in the air for one-two days more and I will post a solution during the weekend unless somebody else does so.
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