matt grime

Doron, all of what I have written doesn't depend on defining the cardinality of N. I do not need to define cardinality, indeed I cannot define cardinal numbers without first defining function, injection and surjection. So, you do not need cardinals to do set theory. I do not need to "find" the cardinality of N, whatever thaat means, to show that f(x)=x is a bijection. "for each" and "for all" are semantically the same in mathematics when we use it about elements of some set. So I cannot prove, define, explore, whatever you prefer to say, that f(x)=x^2 is not a surjection from N to N? even though it isn't since there is no integer that squares to 2? "In order to use the word ‘ALL’ you first have to prove that the cardinality of infinitely many elements CAN BE FOUND." no, that isn't how other people use "all" that is your view on it, and it is wrong (in the sense that it is inconsistent with its use in the rest of the world). "In order to speak about the properties of |N| you first have to define |N|" which part of the sentence where I explain that I'm not stating any result that has anything to do with |N| do you not understand? I don't need to define cardinals, and I don't care too as their definition and proof that they are consistent with, say, ZFC is actually a very delicate matter. Anyway, what on earth does it mean to "find" the cardinality of N? "But you use the Universal quantifier in order to define the cardinality of N." Do I? Please write out the definition of aleph-0 that you seem to think that I've offered. I mean the proper and rigorous one. And please feel free to show in what way it is that I've proved anything about it. I merely have explained how one can use a universal quantifier to write something about a property that is true for all n in N, and you still don't appear to have grasped its meaning. I cannot prove that last thing you've asked because what you have written is meaningless. Please can someon lock this thread? There is no advance to mathematical knowledge from it and Doron is refusing to even countenance that he may need to learn the proper definitions of the terms he is abusing. It is a waste of everyone's time to leave it open.

there is a general formula for this: p sin x + q cos x = k let p=rcost and q = rsint for some numbers r and t, then q/p = tant, so we can find t, and p^2+q^2=r^2 so we can find r, then we've got rcostsinx + rsintcosx=k or rsin(x+t)=k so you can solve for x (note we've found r, t, and k was known) now do it for p=1, q=-1, k=1/2

erm, cardinality has nothing to do with it, Doron. i can define functions, injections and bijections without needing to mention cardinal numbers, in fact i must do since i need to define them before i can define cardinal numbers (sometimes you really do make me laugh, wearily). indeed cardinal numbers are sufficnelty complicated as to mean that we would rather not have to discuss them when we do not need to. So, in what way have i done anything inconsistent with the definition of set, injection, surjection and function? For all means, well, for all, as in "for all n in the natural numbers greater than 9, 2^n > n^3" means that if n is any element in the natural numbers and n is greater than 9, 2^n is greater than n^3, or equivalently, there is no n in the natural numbers greater than 9 that does not have this property. That is all. (you may wnat to try proving that statement, since it is true. So saying for all n in N, there is a succesor in N states exactly the same as "if n in N then n+1 is in N", it doesn't say something about the properties of |N|, or anything of that nature, it says something about the elements of N. that there is no maximal one. It is a short hand, Doron. We could say that 2^9>9^3 and 2^10 > 10^3 and 2^11^11^3 and... or we could say it in one simple sentence. So, are you saying that, if S is any subset of the integers, then the statement: for all s in S, s^2 is strictly non-negative (ie is positive or zero) is true if S is a finite set and undefined if S is an infinite subset? Because that is what you seem to be implying. Or how about: for all G in GRP (the category of groups), then if G is finite the order of eery element in G divides the order of G. If I write a proof of that that doesn't mention how many finite groups there are, does it magically become false in your system when i tell you that there is not a finite number of finite groups. You can wait for that proof all you want, however your proposition makes absoltely no sense in the english language or in mathematics or in logic

"Now by using a Universal quantifier (the term ALL) on N ( where N is a collection of infinitely many finite objects) then if ALL 1 objects are in N, then it implies that +1 (the Successor) cannot be found." This isn#t what we mean when say P(X) is true for all X in N, Doron. Stop being silly. "Can you prove that?" AS what yo've jsut asked us to prove is nonsensical statement then of course we can't. I can prove that the function f(x) =x for all x in N is a bijection: Let y be in N and let x=y, then f(x)=y. wow. If x and y arte in N and f(x)=f(y) then by definition x=y. Thus it is a bijection and it's an infinite set and I've quantified something with a for all. there, that wasn't so hard was it?

Yep, 20 years of refusing to learn what other people mean apparently. INcidentally, I think we can sum up his arguments aganis caridinals in more mathematical terms. the idea appears tyo be that with finite cardinals we may define a larger cardinal as a union of two smaller ones. However is we do this with aleph-0 we still only get aleph-0. This isn't a problem, obviously, unless you are of the unswerving opnion that infinite cardinals must behave like finite ones (oddly doron is the one who frequently accuses us of being inflexible, when it is his opinion that wo'nt alter). There are fininte cardinals that are the union over an index of smaller cardinality of smaller cardinals. these have been studied, and we have the term regular because of it. if doron wishes to show our definition of an injection from N to N is not valid all he must do is show the negation is true. the negation is that there are distinct naturals x, and y such that f(x)=f(y). as that implies x=y contradicting the assupmtion of them being distinct it could prove difficult. however we can simply expect his disproof to be a flat out denial without evidence. INcidentally, doron has never quite seemed to understand the 3 different levels of criciticism i ahve of his "maths" 1. Whatever else, he doesn't make it clear enough until pressed, and even then grudgungly, that he isn't talking about mathematics and using the terms correctly as any one else would understand them. All mathematical statements are imlpicitly made in some system (ie one wher for all has a definite meaning). You may offer new systems for study, offer new definitions, and as long as you are consistent in doing so, no one minds. you may not offer a new definition of some term, and then state that an old theorem is wrong with that new definition as you have changed the hypotheses. this is his error on many occasions. 2. this is one that i do not often use. doron has rejected various premises of mathematics. that is not unreasonable. i happen to think his objections are silly, and his new axioms are too, but that is neither here nor there. 3. if we are looking at a new system of doronic maths then surely it would behove the author to make clear unambiguous statements about it such as defining ANY of the terms he uses. i do not feel he as done any of this. there is also the 4th criticism perhaps that he misrepresents/misunderstands mathematics. he has for instance completely changed his opinions on the ntural numbers - originally he thought they were defined in the axioms of ZFC - they aren't - now he is moving for a set theoretic free idea of peano's axioms. in the face of such shifting it has hard to take seriously any assertions made about extant mathematics.

"Matt, you did not show to us how 'ALL' and '+1' (the successor) do not contradicting e each other" willl you please stop indicating to us that you do not what quantifiers mean. We all get that you don't understand them. "If we force ALL on a collection of infinitely many 1, it means that the successor (+1 expression) cannot be found in this collection." This makes no sense as a piece of english, nor as mathematics, "Your {1/2, 2/3, 3/4, 4/5, 5/6, ... , 3/2} is irrelevant in the Natural numbers case , because the successor (+x) in your example becomes smaller, and this is not the case in the +1 successor of the Natural numbers." again nothing to do with anything - the metric isn't important, we aren't treating these a topological. "So 1-1 mapping between collections with infinitely many elements is a non-sense." Define a map from N to N by: f(x)=x. it is a 1-1 map between two infinite sets. I haven't used any quantifiers. A function is a subset of NxN, this is the diagonal subset {x,x} And I see you#re back to stating, without proof, or motivation that infinite cardinals must have the same properties of finite ones. Of course, you've not defined cardinals yet. Still, as long as you understand yo'ure not talking about mathematics as anyone else understands it then we're ok.

Thanks for cheering me up, you can usually be relied upon for saying something hilariously silly. It bears no relation to mathematics though. Just so we can check, do you now accept that threre are infinite sets with a well ordering which possess first and last elements? (max and min, for any mathematicians who may be reading this) just wondering, cos you keep telling me i'm wrong until you evnetually have to accept that actually i'm not, so how long before you accept that when i explain to you how we use "for all" i'm not trying to trick you or force my opinion onto you, but attempting to explain yet another misapprehension you h ave about mathematics. mind you, as you didn't know what a bijection was yet still felt you could assert things abuot them I doubt it'll be anytime soon.

again you're not using "all" correctly as anyone can see, so what has this to do with mathematics? DOron? In fact god knows how yo'ure using it. the statement "for all n in N, there is an n+1 in N" is exactly the same as the statement "if n in N, then there is n+1 in N" if there is a natural number for which its successor is not a natural number, which one is it? are you making statements about mathematics or abotu *your* special doron-maths? and why are you then taking the internal arithmetic of N to conclude things abuot arithmetic of infinite cardinals which you've not defined. that isn't what happens, and I'd hope you'd've learned that by now.

You should at least point out that you are not using the convention definition of 10, 2, 1 or / in that example.

Usually the symbols for the empty set, whilst looking like phi's are not actually phi

4) can also be written: for all n in N, there is an n+1, a successor in N that's just another way of writing it. can you demonstrate that you must use universal quantifiers in Cantor's argument, as that is you "issue" with it. and thus if i were to rewrite it without universal qunatifiers, then you'd suddenly accept it as true within your system? of course that is all pointless since cantor's theorems are not written in the langauge of your system, whatever that maybe. In cases like this we do not usually say that a theorem is wrong, but that it has no analogue. ie in your doron-maths, there is no analogue of infinite cardinals (incidentally, have yuo defined cardinals yet? don't think you have!) and the other requests to explain what on earth the "infinite" "sum" you seem to think |N| is....? incidentally, doesn't the demonstration that there are infinte (well ordered discrete) sets with maximal and minimal elements (something that you said couldn't happen) indicate you ought to tone it down a little? of course it is unclear whether you meant your statement as one about mathematics or your peculiar doron-maths.

Please, DOron, attempt to define well ordered (for N) without using a universal quantifier (for an infinite set), and if you wouldn't mind posting, say, Peano's axioms with proof that they are not universally quantified too (for instance, the induction axiom is universally quantified). And your example of N and epsilon is still nonsense - you've not even defined epsilon to the point where anyone knows what you're talking about. and as another example you state that we cannot calculate the sum of infinitely many elements. No one has caclulated any such (algebraic we presume?) sum. What sum?

Here we are, page 1 of this very thread. Doron, you may work with a differenet definition for the words, perhaps I misspoke to say you shouldn't: there are obviously different meanings for the same word dependent upong context even in maths. However, if you are going to use a different and contradictory meaning then you must clearly say so as the first thing you define, so as not to confuse anyone into thinking that you're talking about their use of the word. At least we can clearly see that you admit you aren't doing mathematics as others do, so we can leave you to your world where "complete" means "finite", which means "first and last element", in which sets have "next elements" but are'nt well ordered apparently, and which contains all other misuses of words. Do you remember the analogy I gave of soemone going to France and refusing to speak French because they thought the meaning of words should be open to change by anyone? Well, it's not a great analogy because words do change their meanine, but not because one person wanders into a foreign country and tells them they're all wrong.

That isn't the sense in which we use the word in mathematics. (Definitions are fixed after they have been agreed upon, Doron, you're moving the goal posts again.) This isn't being immodest, it is knowing the accepted meanings of the words. Moreover you are implicitly assuming that all sets are well ordered, and I can show you how to well order every (only) set if we assume the axiom of choice is true. This also highlights what I'm trying to get across to you (and any interested onlookers): you are perfectly free to develop whatever theory you want, and prove whatever results are consistent with it, however you cannot say: Cantor is rubbish because he uses "all" for an infinite set. When you are using, as we can clearly see you have stated, the word in a completely different sense. At best Cantor is "not true" in your system, because you have picked a different system; it is on surprise that one has different results if one changes the hypotheses. Cantor didn't make his statements about a system such as yours, that is all. Stop. moving the goal posts.

This just shows that you do not understand what the word "all" means in mathematics, that's, erm, all. If you don#'t like it then use its negation.

"In a collection of infinitely many elements, an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many alamants) that cannot give us the ability to force a universal quantification on this kind of a collection." the words epsilon, invariant, next, are undefined, and the conclusion is mathematically unsound in the proper world of mathematics. Tell, you what, why don't you offer a theorem that we know to be true in mathematics and show it is false within that system. Not within your system where none of the terms are defined properly. Since you are talking about cardinality we can only presume you mean in proper mathematics not in your ill-informed mind. So we can only answer in proper mathematical terms.

|N| means, approximately, the class of all sets that are in bijection with N. There is no issue there apart from your ignorance of the meaning of bijection and refusal to accept that inversal qunatifiers are usable with infinite sets. That isn't a problem with mathematics, but with you. So please, go away and develop a mathematical system without universal quantifiers, in which one cannot talk about functions between infinite sets and leave maths alone - it is, so far, consistent in itself.

You mean apart from the fact that peano's axioms are usually cosntructed by sets? N, the natural numbers is a model of w, the first infinite ordinal. You do understand that? w is a (an equivalence class of) set(s): a well ordered countable set with no maximal element, it is naturally isomorphic to N with its usual ordering.

I'll give you an example of a well ordered set with infintely many elements and a maximal element: the ordinals w+1, w+2,.. w+r..., pw+r,.. are all well ordered sets with a maximal element. A model of one of them in the reals would be {1/2,2/3,4/5.5/6,...} u{3/2} interestingly you can use infinite ordinals such as these and transfinite induction to show that a continuous function on a closed boudned interval is bounded. you might want to look up the definition of continuous Doron (or function for that matter) since you appear not to know what that means either.

Note, we're picking N first, right, so i can take a new sequence y_m where y_r=x_{N+r} then that sequence has the property that for all e>0 and for all n |y_n_x|<e

Or if you prefer you know that after N evey element of x_n is in EVERY neighbourhood of x, but the only way for that to happen is for all the x_n after the N'th to be equal to X. Note it is for ALL e>0

Eventually constant. If x_n is not eventually constant at x, then i can pick an infinite subsequence y_m of the x_n such that non of the y_n are equal to x: if it isn't constant I can find a first place where x_n differs, and then a second and so on... (so I can assume that non of the x_n are equal to x by passing to a subsequence if i wihsed) Let N be any integer, and let e=|y_{N+1}-x|/2 Then for this N I know that the assertion: for all e>0 and for all n>N |y_n-x|<e is false. but this N was completely arbitrary so i've shown that if the sequence is not eventually constant at x then the condition is false. or equivalently if the codnition is true then the sequence is eventually constant at x.

Yes, so the idea that the first and second are the same we can clearly see is wrong by this example. At the risk of simply repeating myself, the secoond one states that given a sequence x_n and some number x that we can find AN integer N such that for all e>0 and all n>N we have that |x_n-x|<e Consider the negation instead if that helps: for all N there is an e>0 and an n>N such that |x_n-x| => e. this implies the sequence is not eventually constant and equal to x since it states that no matter what N is ,there is a term after the N'th which is some positive distance from x. that is the sequence is not after any point always equal to x, agreed?

But what about infinte sets that have maximal and minimal elements, Doron? Oh, please define cardinal too. Bloodhound is a second year undergraduate, I can forgive him his errors. People who claim to have earth shattering views on mathematics that are plain silly can't be forgiven for repeating mistakes. Not least since it is immaterial whether you think there are infinte sets or not. If you disallow them then it's your problem - we have them and they cause no problem to us, just interesting questions. Oh, and yo'uve not defined a set either. Clearly it can't be a collection of things, since that allows for infinite sets and here you say: "But then we see that is we use a universal quntification, then our set does not exist or our set is a finite set" of course, it may just be that you don't understand universal quantification...