matt grime

Thinking out loud: let R be the ring, I an ideal of R. Let p be an element of minimal degree. Either p generates I and we are done, or there is some element in I not in IR. Obivously, no element in I\I(p) can have degree deg(p), otherwise if q were one then we could choose t in k such that p-tq is an element of deg less than deg p or deg q. So, let q be some element in I\I(p) of smallest degree, then p and q must generate I: suppose r were an element not in the span of p and q, with minimal degree, then we can manufacture an element from combinations of p, q, x^2 and x^3 having the same degree and leading coefficient, call it s, and then r-s would be in I, and not in the ideal generated by p and q, and of smaller degree than r, so it is zero.

Doron, maths does not behave as if a 5 year old needs to understand it. 1. define a set, 2. define a map of sets, we'll assume we all have the same notion of these shall we? f is a bijection if f is invertible.

Yes, it's a very common problem that people do when they first do combinatorics (counting proofs) binomial coeffs have the following properties: [math]\sum_{r=0}^{n}\binom{n}{r} = 2^n[/math] [math] \binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r}[/math] [math]\binom{p}{r}[/math] is divisible by p if p is a prime and r is between 1 and p-1. they are all amongst some of the first things you might be asked to show abuot them, and there are many many more relations.

I think the point is that you've not given sufficient information to uniquely determine the function. You need to state what the input set of values is. The function f(x) = 2^x - 1 + sin(2xpi) also satisfies the constraints yo'uve given, amongst infintely many others.

I did, it makes no sense at all. Demonstrate why the map f(z)=z-1 is not a bijection from Z to Z. INcidentally, here is the proof that it is a bijection: let g(z)=z+1, then fg(z)=gf(z)=z, hence f is invertible, and thus a bijection. Where is this wrong? For any moderators: If Doron cannot do this, please can you lock the thread again. We have defined bijection for him, and he has not acknowledged the definition, nor has he shown why simple maps that we can see are bijections, cannot be bijections. The introduction of cardinals is completely spurious and is yet another attempt to distract from the fact that he doesn't know what he is talking about by dressing it up in more pointless complicated waffle.

Actually, you're contradicting yourself here too. You claim that you cannot quantify for all for infinite sets. Yet you're saying, from one example that has nothing to do with the matter in hand that all maps between all infinite sets are not bijections. How can you say that? YOu've jsut qunatified an infinite set with a for all.

Doron, this has nothing to do with anything. Cantor's argument is in essence about the fact that there is no bijection between N and its powerset P(N). It the does not define a beijction, it shows that one cannot exist, exactly as we want. It is nothing to do with whether or not f(z)=z-1 is a bijection from Z to Z. If it isn't a bijection, as you claim it cannot be, then demonstrate that it isn't a biejction, that it isn't invertible.

A stupid one? Who on earth says maths must be understandable by 5 year olds? Mind you I suspect a 5 year old might be able to spot the holes in Doron's arguments. We have defined a bijection in two simpel and equivalent ways. I can add a third if you like that generalizes to the notion of isomoprhism which is an invertible arrow in a category. As I say, you claim to have spotted an egregious error in the definition, so one presumes you already know the definition. So, is the map fomr the ring of integers to itself given by f(z)=z-1 a bijection? If not why not, since every one else can see it is a bijection.

look one post up.

equivalently a map is a beijction between sets if and only if it possesses an inverse. As you've said the bijections are fundamentally wrong (as are cardinals) why am I explaining this to you? surely you know the definition (otherwise, how can you have formed an opinion). nb, the inverse of a set function f from X to Y is a map g from Y to X statisfying gf(x)=x, and fg(y)=y whenever x is an element of X and y is an element of Y.

Is it a bijection? A simple yes or no is required, though a proof would be handy.

what do you mean by next then? so are you saying that that function is a bijection?

"The existence of the next" who on earth is claiming that all sets can be well ordered? (they can if we assumu the axiom of choice, but so what)? No one else apart from you is talking about successors. They are not important in the argument. So, Let S be any set, and define f:S-> S by f(x)=x. Show where, with the proper definition of all the terms, this is wrong.

you need the inclusion-exclusion principle: you did the number of permutations, then subtracted the number with two books together, now add back the number with three books together (and subtract that with 4 books together)

so far doron, there exists exactly one person who thinks the word quantity is what we're talking about. and he/she hasn't defined it mathematically. would you care to correct that? and what, pray tell do you mean by exact? or cardinal? or ALL or related or exist? your paragraph is once more the work of an illiterate. what successor? apart from you (a person of very limited mathematical sophistication it has to be said) who has talked about succesors? again, no one has mentioned the word successor in the context of defining cardinals of infinite sets... the successor of N? in what way did Cantor define such a thing? he didn't use the phrase "shared property" either, nor did he talk about "infinitely many N members". he used a the axiom of choice to show that there in we may totally order the infinite cardinals, but what's that to with this argument? not a lot as the ordering isn't defined in terms of the successors of peano's postulates... are you about to define those terms.... oh, that'd be a no then... let us omit some of the junk... ahem, so apart from not defining epsilon properly, and once more talking about the VALUE of|N| what is going on here? nothing apart from the ravings of yet another ignorant crackpot.... what on earth do you mean by finding the value of |N|? that is yet another piece of nonsense and doen't even begin to explain why the map from a set S to itself as given by f(x)=x isn't a bijection. So, go on , Doron, please explain why f(x)=x isn't a bijection from S to S. please, pretty please..... And could you at some point explain to the ignroant masses why defining |N| as the iso class of N in SET isn't sound?

doron, i am saying nothing abuot all 1-1 maps, nor am i saying anything abuot checking all of them, or "finding" all of them. i have defined |N|, it is the isomorphism class of N in the category SET. please note that i corrected part 1. in my psot to FINITE, as it should read, rather than INFINITE, which is obviously wrong: we are attempting to define a reasonalbe notion that some may call (mistakenly, in my view) size for infinite sets.

doron, that is nonsense, utter tripe. look up the definitions. read them, understand them. try getting someone to translate to english for you. i am saying nothing about the "complete collection of 1-1 mapping", at least i don't think i am, since that phrase is meaningless.

Read it carefully Doron, Cantor noted that 1. we have a clear idea of what we mean by size for finite sets. 2. this could be summarized by : two finite sets have the same size if and only if there is a bijection between them. 3. that this equivalent notion does not mention whether or not the sets are finite. 4. that we can thus use this GENERALIZATION of the notion of size of finite sets to distinguish between infinite sets in an ANALOGOUS manner. 5. Initially we say that two sets gave the same cardinality iff there is a bijection between them. Using this, and using analogy, we say things like N is smaller than R, but this is just stating that there is no bijection between them. These are noty pyhsical objects, they do not have mass, width or breadth, this usage of the word size, or smaller or such is a convention that is a generalization of the finite case. It does not contradict the finite case and is reasonable to most people. 6. starting with this notion of cardinality we ar led to invent cardinal numbers. This is a generaliztion of the order of a finite set. after all we say that {1,2,3} and {2,3,4} have the same size, three elements, in the finite case, so why should we not generalize this to the infinite case and pick some special representative of the class of all sets that are in bijection with N? we do and call this aleph-0, so when we say |S| is aleph-0, which we shall explain in a second, we mean exactly and no more than there is a bijecton with the natural numbers. 8 having decided that we can use infinite cardinals, cantor showed that, with the axiom of choice the cardinals are well ordered, hence the aleph-0, aleph-1 and so on representatives of the equivlance classes of sets modulo bijection. 9. conway then showed how to put an algebraic structure on them. 10 when we say aleph-0+1 = aleph-0 then we are saying nothing more than a statement about the existence of maps between the union fo two sets and another. 9 sum, size, number as you understand them for the finite case do not have any association to the infinite case a priori. we have generalized them. consequently there are differences between the behaviour in the infinite and finite cases. this is not a problem.

Oh dear God. DEFINITION avoiding the use of all. F:X --> Y is a surjection if for each y in Y, there is some x in X with f(x)=y defintion of injection, f is an injection if f(x)=f(y) implies x=y. Define a map from N to N, or ANY set to itself by f(x)=x, this is clearly a function, since it is the relation defined by (x,x), the diagonal subset. I;ve not even used the word all once have i? now, f is a bijection since, give y in N, let x=y, and then f(x)=y so it is a surjection, and f(x)=f(y) imlpies x=y, so it is an injection. thus it is a bijection. It;s the sodding definitions.

OK, so by your definition "sum" cannot be used for an infinite collection. This is not unusual, but so what? You're using, so far, the words sum, cardinality, size in a way no one else agrees with, in this thread. there are two options, at least, 1. everyone else is wrong in some absolute sense, and you are right, the whole of mathematics is ill-founded and based upon wrong ideas and we are all trying to suppress your iconoclastic work or, and i'm leaning towards this one to be honest, 2. you know nothing about mathematics, and don't understand the idea of axiomatization. (based on, amongst other things, the fact that f(x)=x can sometimes not be a bijection..) you are perfectly entitled to take other axioms, but you are not entitled to say we are wrong about them since there is no absolute truth in these issues. you may say that size is something that cannot be applied to infinte sets, or that sum cannot be used. however, no decent mathematician actually uses either of those terms in the way you imply we do. if we do say size it is an illustration only and if pressed we fall back on the proper definition. now, moderators, can we please lock off this thread too? or shall we go through the while pointless routine whereby a crank (this legally is not libellous in the US by the way) tells mathematicians that he understands mathematics better than they do, and the rebuttals which point out that the allegations are neither good english or mathematics, and whilst you may certainly offer a new sustem of mathematics, you ar in no position ot make judgements about the proper, for want of a better word, mathematics are ignored.

Again, that is not the definition of cardinality. again, that makes no sense in any form. this isn't a mathematical statement this is a non sequitur. as one is the 'negation' of the other...? who on earth is concluding anything about the "complete quantity"other than that the set is not finite? we are not finding any sums, though. we are not adding anything up. again, all and complete are not perfect synonyms in english, and even if they were it isn't the common english meaning that is important. complete has several meanings in mathematics, such as all cauchy sequences have a limit, or all polynomials possess all their roots. and this isn't even a matter of "all" it is a matter of "for all". ahem, you aren't acknowledging that the *definition* that two sets have the same cardinality means that there is a bijection between them. the notion of summing the elements of an arbitrary set doesn't even make sense without further explanation. "claim", clime means weather or weather conditions do you even know the definition of injection? cardinals (infinite ones) belong to strictly higher system than functions since we need to define bijection (or injection by the bernstein-cantor-schroeder theorem) in order to define cardinals for all sets. excuse me? what is the cardinality of a function? i can see how one may choose to define it, but it is always equal to the cardinality of the domain so it is pointless. again this is an undefined use of the word sum. more meaningless waffle, I'm afraid. Here's a simple statement: [math] (\forall n \in \mathbb{N})((n>9) \implies (2^n > n^3))[/math] in what way have I done something that is *mathematically* wrong? Not in your opinion that no one else understands, what have I don'e wrong given the correct(ie the ones everyone else uses) meanings of all the terms?