matt grime

I didn't even notice 'interative' v 'iterative'. Find me one person who has said infinity is the result of an iterative process. "The Flat Pitch of Number Theory Becomes Curved." what does that mean? "So if we ask how many Numbers are there in the Real Set ? We say there are Finitely Realizable Number points." No, we shall not say this. the set of natural numbers is infinite. Any finite set of numbers has a greatest elemen, what is yours? You're talking nonsense, and confusing 'real life' with maths. sure we're only ever going to use a finite number of numbers, but that doesn't mean you can declare the set of all possible numbers to be finite. they aren't a concrete set of things you're going to stub your toe on, they are a concept, an idea, and they are an infinite set. you ain't going to convince anyone otherwise, not least if you keep asserting that 'infinity is the result of an iterative process' which no one else appears to have done, and no one understands. when i said i can't offer a simple definition of infinity and that i suspect there might not be such a thing i was strictly refering to the real world around us. there are certainly infinite sets in maths, infinite cardinals, sums from one to infinity, points at infinity.

But that isn't between 0.99... and 1 and has nothing to do with anything. Stop the necrophilia.

since these are dummy variables let's keep the names separate. so set y=-x, and dy=-dx then the upper limit of x=b must mean y=-x=-b, and the lower limit becomes -a.

1. Work out all 5C3 possibilities. It's not hard, just tedious. 2. Appeal to the pigeonhole principle. needs a little ingenuity.

Oh. My. God. I feel the compelling urge to correct everything in sight based upon one sentence. "Algebra is a branch of mathematics concerned with structure and quantity, but not with arithmetic because it uses symbols rather than numbers." how do i find out who wrote that?

Absolutely, the tree. The only subjectivity involved is to do with people choosing the best model (for whatever criteria they see fit) and deciding on what level of approximation they want to use when writing out any numerical output that may be produced. The maths and the conclusions drawn purely about the mathematics are not subjective; there is nothing to interpret in and of itself. One subjective thing to do with this is how much anything of this nature has to do with mathematics at all.

what on earth is an interative process? if you don't define your terms properly they cannot be 'disproven' or 'proven'. they just make no sense. i will not even attempt to disprove that since I do not think 'infinity is an interative process', nor do i think its negation is true since i have no idea what that even means. point out one person that has said infinity is an interative process. or a process fullstop. i can tell you how infinity is used in maths (many ways) and i will keep repeating that it is merely a short hand to indicate 'not finite' in different ways, and any statement involving the word can almost certainly be rewritten to omit the word entirely. i can't offer a simple 'definition' of what infinity 'is', since i don't necessarily think there is one, or needs to be one *for mathematics*. (note i didn't contradict myself, i pointed out that mathematicians are fine with infinty and what we mean when we use it, but that non-mathematicians by dint of doing weird ill-defined hand wavy things without explaining themselves clearly are not. this thread is a perfect example of that confusion. give me any instance of 'infinity' being used in mathematics and i can explain it. you're not talking about maths here, hence the suggestion you try a different forum.)

That might well work, but for the newbies remember a lot of tex stuff is deprecated in latex2e (documentclass for instance doesn't work in tex).

It's only a paradox because you think it is unlikely, when it isn't. If there were 183 or more people in the room then the probability that there are two people with consecutuve birthdays is 1. If you have 23 people in a room then there is a common birthday with probability greater than 1/2.

it's a slash followed by a space. eg "\ " without the quotation marks. You can insert larger spaces. Use google to get the information; there is a lot of latex on line (no puns please).

You cannot prove a theory that is entirely subjective. Nor did you even explain what any of those terms actually mean' date=' ie what makes something observer independent or otherwise. Try giving a reference where someone actually claims that maths is observer independent as a science. If by 'we' you mean the general public and the belief that somehow mathematical things are 'real' then they don't. However, in maths, the symbol [math]\infty[/math] and its uses, as well as other things with the label infinity are prefectly well understood as 'not finite'. It has different context dependent meanings, as do many words and symbols. Sorry, you've stepped fully into the realm of crackpottery, or at least philosophy. I won't respond to the rest of the post. It is not mathematics. Try the speculations section. Ok, I admit, crackpottery is a tad harsh, but it has the hall marks of many a mathematical pet theory that has no relation to mathematics. Perhaps you're just posting in the wrong place.

You've note defined 'number' still, yet keep talking about them as they have some well known definition that you're using. You don't mean 'verb' you mean 'noun', and you got back a definition. The cardinality of R is c. That is the symbol we use to describe its cardinality. Just as I use the symbol 3 to describe the cardinality of {a,b,c} (assuming they are all distinct). All sets that are bijective with R also have cardinality labelled c. With the axiom of choice it is possible to well order cardinals so that given two sets either |S|<|T|, |S|>|T| or |S|=|T|. As i said, roughly speaking we can think of cardinals as being equivalence classes of sets. After all, what is 3? You demonstrate 3 to a child be counting off 1, then two then 3. It's actually quite deep really. We can unambiguously state when some set has 3 elements by declaring it to be so if it is in bijection with some canonical set with 3 elements. See eg the peano axioms. A group is a group is a group. There are simpler algebraic objects, and more complicated ones. It is not 'incomplete' as a definition. It might not do what you want but that is a problem with you misusing something. Get a better tool.

For the tree, the rationals in latex are \mathbb{Q} note that the reason for this 'problem' is that both of you are using operations that are not the group operation to obtain something, so it is no surprise closure fails since no one is saying that the reals are a group with respect to multiplication, and square roots aren't even binary operations.

The real numbers are only a group under addition. No finite number of additions of real numbers yields infinity as an answer. It cannot or it would not be a group. You write the operation multiuplicatively. The reals are not a group under multiplication. The nonzero reals are. again no finite product of nonzero real numbers is infinity. 'infinity' as you are talking about is purely an analytic thing and merely is short hand to describe behaviour *that is not finite*. It is unwise to use infinity as a noun in this way for precisely the misapprehensions it has created in your understanding. What does the phrase [math] \lim_{x \to 0} 1/x = \infty[/math] actually mean? I can write it our without ever using the symbol [math]\infty[/math] or the word infinity. It is equivalent to the statement, for all e>0 and all L in R, there is an x such that 0<|x|<e and |x|>L. See, I didn't mention the word infinity once. Further, I can replace this infinite sum [math]\sum_{n=1}^{\infty}x_n[/math] with one that does not use the symbol or word infinty as well [math]\sum_{n\in \mathbb{N}} x_n[/math] Nothing holds us back from 'accepting' infinity, or more properly 'using symbols that are in some sense larger than any real number' for algebriac operations, they are just not elements of the set of Real numbers. Just as we can extend the reals to allow i, and get the complex numbers. Incidentally the term 'real' in real numbers in no way is supposed to imply that these are 'real' in the ordinary language use of the word. Look up non-standard analysis or hyperreal numbers. Also try the extend real line. By defintion 'the size of a set' is called its cardinality. Look up transfinite numbers or infinite cardinals. I already gave you two examples, c, and aleph-1, which may or may not be the same. Aleph-0 is the cardinality of the set of natural numbers. Approximately a cardinal is the class of isomorphic sets. "Infinity(whether good or bad) comes as a consequence of its existence" is in my opinion nothing to do with maths. I can't even decide what that means, if it means anything.

Or in general you appear to be saying that x+1+x+1=x+2

no' date=' as you can tell where i said that i is an element of an extension of R in my first reply to you, and therefore asked you to explain what criteria you were using to say things are and aren't numbers. No it isn't. This again is why i wanted you to state what you meant by 'number' if you were going to make any claims about them. is that your definition of 'universal set'? all the measurable quantities of what? the cardinality of the real numbers is 'a measure of quantity' and is not a real number. it's (the set of real numbers) cardinality is c, the cardinality of the continuum. according to the continuum hypothesis (which is known to be independent of ZF) this either is or is not the same as aleph-1, the first uncountable cardinal. Only if you're assuming that the reals are 'a universal set'. I've no idea why you'd assume that or even what your definition of a universal set is. Generally maths does not operate in a set theory that allows for a 'universal' set, ie the set of all sets (this is not the same as presuming that our sets exist in some universe though). but i doubt you're using any of these terms with the accepted standard meanings, given that you believe there is an infinite cardinal in the set of real numbers. I can give you the definitions of what the real numbers are. that should be a solid enough reason. this isn't about 'opinion', dkv, but about the deductions from the definitions. R, the set of reals, is the completion of the rationals in the euclidean norm, it is the set of dedekind cuts of Q, it is the totally ordered field. it does not contain 'infinity' as an element (there is no cauchy sequence of rationals 'converging to infinity'. it is not an element of R. if you think it is then you are simply mistaken. A model for the set of reals is the set of decimal expansions, so that the reals are numbers of the form [iNTEGER PART].{fractional part} (modulo the relation that we identify certain strings such as 0.9.... and 1) and obviously given any real number there is an integer [iNTEGER PART]+1 that is bigger than it. Even if you didn't see it before surely now you can see that there is no such real number as 'infinity' if given any real number we can find an integer larger than it. The cardinality of any infinite set is not a finite number and not in the set of real numbers.

Define the plus of S(x)+S(x) But I would say no, since what you've written is, with the 'best interpretation' saying that 2+2=3.

2 is, by definition, the result of adding 1 to 1 when we're dealing with numbers as abstract objects. Peano's axioms give a way of 'concretely' realizing numbers as a collection of objects labelled 1,2,3,... and an operation, 'addition', such that adding two objects labelled 1 gives the object labelled 2, and further that all the other properties of addition hold. Let me use [n] to mean the object labelled by n, then you need to check that in the Peano system that doing ([n]+[m])+[p] is the same as doing [n]+([m]+[p]), and that [n]+[m]=[m]+[n]. I will say that [n] is a collection of sets of sets (of sets of sets) and addition is something like taking the union, so you do need to check that these constructions behave properly. I don't propose to give it here since it is notationally heavy, I'll only get it wrong, and there are plenty of other places where it is explained properly. Thus we're saying that there is something relatively 'concrete' that fits our abstract set of rules for the natural numbers. Some people find this to be reassuring. There are results about the natural numbers as an abstract object that cannot be deduced from the Peano Axioms alone (try googling mathworld peano to find out more on this).

if you mean 'an element' by 'part', then of course not. if you meant is there some part of the properties of the real numbers that is infinite then of course there is. what you wrote doesn't let one decide if you're incorrect or speaking about something else. nope, that doesn't make mathematical sense. no it is not self referring that doesn't mean infinity is an element of the set of real numbers. *sigh* 0 represents the additive identity, whatever label you choose for the additive identity (and they are just labels) 0/0 does not make sense. you cannot change the meaning of a symbol and then claim that it solves the original problem.

It in now way depends upon the size of the observer. 1km is 1km. There is a difference between 'infinite' and 'being too big to contemplate'. One is mathematics, the other is your subjective opinion.

Infinity is not part of the real number system in the sense I gave. Point out one reference that states it is. Since you are simply using it for limits and induction, then it is clear that you aren't actually using infinity as a 'number' (which you didn't appear to define; your writing is hard to read if you don't use breaks), merely as the quality of not being finite. Which is good, and all it should be used for really, but that doesn't make it 'part of the real number system' in the sense I used of 'being a real number'. If you are going to use infinte cardinals then there is a proper class of them. That is why it is ambiguous. You talk of infinity as a 'number' without saying how you're using it. i as the square root of one is not nonsensical at all. It is merely difficult for people to accept because of the nonsense they were taught in schools about maths. Look at the historical analogy of the (probably apocryphal) murder of the person who proved that the square root of 2 was irrational. Surely you accept sqrt(2) is fine, so why is i nonsensical? Because if you square a real number it becomes positive? Sure, that just states that you can't take square roots of negative reals and get a real, but that doesn't mean that there is no such way to extend square roots to accept negative arguments. Just as we shouldn't restrict to square rooting only numbers that give rational answers. There is no consistent way to extend the fractional notation of a/b to accept 0/0, that is why it is genuinely meaningless in mathematics. If you have to resort to arguments about apples then you're either insulting my intelligence or your own. Stick to talking mathematically when discussing maths.

The answer can't be 20. Any way of pairing these people up must leave one person out on their own. Their are 9 different ways to pick one person to leave out, and so the answer must be divisible by 9. Anyway, counting now, we can leave out1 person in nine ways, and you need to figure out how many ways to split 8 people into pairs. Just do it with a pen and paper. Perhaps try doing it for 4 people, first, then 6 poeple, try to spot a pattern.

we have a distinction already between computable and non-computable numbers. strictly speaking i is not in R, but in an extension of R, so you need to explain what you consider a number to be exactly before you start dismissing things as meaningless examples of them. Incidentally, all known discrete models of space are 'inaccurate': they make physical predictions that are observably false, so it is a little premature to say everything happens in a quantized way.

Given a matrix A there is not always in inverse matrix. Sorry if you didn't mean that, but that's the implication I get from your post. I think googling for the adjugate matrix will help you.

surely the warning that the string is too long explains why it won't work on this forum. If you want to see it just latex it at home using your installation of latex. This is freely available for all *nix platforms, windows, mac OSX, and probably many more besides. just download the relevant version and enjoy. use google to find it.