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From: Tony Orlow on 31 Oct 2005 13:24 Randy Poe said: > > Tony Orlow wrote: > > David R Tribble said: > > > We are forced to conclude that there is no natural s that maps to > > > *N, and that therefore your mapping scheme is not a bijection > > > between *N and P(*N). > > You are not forced to conclude that this prevents a bijection. The entire > > infinite set would be the final subset enumerated, but the set does not end, > > nor does the set of its subsets, so the bijection will never, ever get to this > > point. You used "TOmatics" fairly well. What if we apply it to the evens? > > Whatever n you choose, you have 2n, but 2n is a natural too, so you have to map > > it to 4n, etc. > > The objection above is that there exists at least one element > of P(*N) which is not equal to f(s) for any s in *N. That > objection says nothing about "last". Ahem. Is the entire set *N not the last element in the ordered set P(*N)? In my biojection it is, and it's the entire set is generally considered the last element of the power set, the null set being the first. So, at least in the case of my bijection, this is exactly what that missing mapping translates to, and it is probably provably always the case. > > The same objection can not be raised about f:N->E, f(x) = 2x. > Every element in E is mapped by some element in N. Every > single one. Nobody but you says you have to consider "the > last" in order to justify the claim that this statement is > true for every single member of the set of evens. But you are saying that for my bijection, effectively. COntradiction abound by the unspoken assumptions of such last elements in this transfinite mess. > > The surjection part of my simple proof of the bijectivity of > f(x)=2x had this simple structure: Let y be any element of the > codomain. Then there exists x such that f(x) = y. > > If you wanted to establish surjectivity of your map f:*N->P(*N) > you need the same structure: Let y be any element of P(*N). > Then (TO can show) there exists x in *N such that f(*N) = y. That bijection is established by the mutual bijection between the infinite binary strings and these two sets. If Randy can just "say" that *N is the same as the set of all infinite binary strings, then why can't I? And, certainly, if I am representing the power set as a set of binary strings, there is a bit for every one of the infinite number of naturals in *N, so it is also the set of infinite binary strings. So, what is your objection? What is left to be proven? > > But of course that last part isn't possible. Because we > know at least one example of a y in P(*N) such that > x does NOT exist. Yeah, right, the whole enitre unending set with no last element, which corresponds to the unending string of 1's with no last 1. That doesn't count, Mr. Largest Finite. > > Your claim that surjective proofs are impossible without > identifying a last element is pure nonsense. Where is this > "last element" in my now-TO-approved proof of the bijectivity > of f(x) = 2x? I made no such claim. Again you miss the point. You are the one deriving a contradiction by asking for the last element of the infinite set P(*N), which is what *N is. It doesn't exist in any completed form. It is unending. But you declare the end and want a mapping. The contradiction is in the question, not the answer. > > - Randy > > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Tony Orlow on 31 Oct 2005 13:28 Virgil said: > > Albrecht Storz wrote: > > >> My argumentation is very easy: > > >> Every nat. number represents a set. If you look at the first 100 nat. > > >> numbers, the 100th nat. number "100" represents the set {1, ... , 100}. > > >> As this holds for every nat. number, if there are infinite nat. numbers > > >> there must be a infiniteth nat. number representing this set. > > Albrecht is arguing that if every A is a B, then it must be the case > that every B is also an A. No, dumbass, he is arguing that if A=B, then if A is finite, B is finite, and if A is infinite B is infinite, which is perfectly true, despite your whining. > > That every natural represents a set does not imply that every set > represents a natural. > > Albrecht would fail any logic class with such a foolish claim. But I bet he gets the girls. ;) > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Tony Orlow on 31 Oct 2005 13:29 albstorz(a)gmx.de said: > > David R Tribble wrote: > > > > > Consider the set of reals in the interval [0,1], that is, the set > > S = {x in R : 0 <= x <= 1}. The elements of this set cannot be > > enumerated by the naturals (which is why it is called an "uncountably > > infinite" set). But all sets have a size, so this set must have a > > size that is not a natural number. It is meaningless (and just > > plain false) to say this set "has no size" or "is not a set". > > > I'm not shure if the reals build a set in spite of you and Cantor and > others are shure. > A set is defined by consisting of discrete, distinguishable, individual > elements. Now tell me: what separates a point on a line from the very > next point on the line to be discrete? What separates sqrt(2) from the > very next real number to be discrete? > If you look only on individual points, you may have a set. But if you > look on all of them? > > So, your above argumentation has no relevance to me. Proof the reals to > be a set, then let's talk again. > > Regards > > AS > > Well, Albrecht, perhaps you should click on the link below and see the Well Ordering of the Real Numbers. I think it's pretty snazzy! Enjoy! -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Tony Orlow on 31 Oct 2005 13:42 albstorz(a)gmx.de said: > David R Tribble wrote: > > David R Tribble said: > > >> So I'm giving you set S, which obviously does not contain any > > >> infinite numbers. So by your rule, the set is finite, right? > > > > > > > Tony Orlow wrote: > > >> If it doesn't contain any infinite members, it's not infinite. > > >> There is no way you have an infinite number of them > > >> without achieving infinite values within the set. > > > > > > > David R Tribble wrote: > > >> Yes, you and Albrecht keep saying that repeatedly. Please demonstrate > > >> why it must be so, because it's not. > > > > > > > Albrect Storz wrote: > > > Your argumentation is not fair, but I don't wonder about that. > > > _You_ has to show, that in the case of the whole set there is no > > > natural number as big as the whole set. > > > > No, I don't, since this has been established mathematics for well > > over a hundred years. You are making claims that contradict > > established mathematics, so it is your task to support your claims > > with proof. > > Only one hundred years. What's that in relation to infinity? > > > > > > But anyway, it is trivially easy to show that no natural can represent > > the size of the set of naturals: > > > > Let N+ = {1,2,3,...}, the set of finite naturals greater than zero. > > (We'll use N+ instead of N because it seems to appeal to your sense > > of logic.) > > I don't know N+. I suppose you consider N instead of N_0. > > > > > > Suppose m in N+ represents the number of elements in N+. Since m > > is a member of N+, it must be a finite natural. Since m represents > > the number of elements in N+, each of which enumerates itself, m > > must be greater than all the other members in N+. But every finite > > natural has a finite natural successor, so m+1 must exist, and m+1 > > must also be a finite natural, and thus a member of N+. But m+1 is > > greater than m, which contradicts our supposition that m is greater > > than all other members in N+. Contradiction. > > > > This forces us to conclude that we cannot find any m in set N+ that > > represents the size of N+. Which in turn forces us to conclude > > that the size of set N+ is not a natural number. QED. > > Define "size". > If size means anything to sets it should mean the number of elements in > the set. If there are elements in the set, there should be a natural > number which describe the size of the set. I agree. > Now, if there are elements in the set, but they are uncountable many, > the word "size" could only be understand in a different way than in the > finite case. > So you are right. Unless you allow infinite naturals with an infinite number of digits, which is quite possible. > (Assuming that a collection of infinite things is sensful to depict as > a set since you can't construct an infinite set, you only can declare > it. There is no connection betweeen every "regular set" and your > infinite sets.) Truly infinite sets come in two varieties, discrete and continuous. Discrete infinities are generated by recursive definitions. > > > > > > > > > You argue: there is no infinite natural number since the peano axioms > > > don't allow an infinite natural number. > > > That's right. I agree with you. > > > > You seem to be saying that infinite naturals cannot exist. > > If you really are saying this, then apologies to you. > > (But no apologies to Tony, who does not believe this.) > > There is no infinite natural number since infinity is unconstructable, > uncountable, unexplorable, unnameable, unapproachable, etc. Bah! Come to my end of the number circle and see how it looks from here. Now there are no finites, but I know they are there. > > > > > > > > > But that's no proof about sets. That's only an aspect of the definition > > > which contradicts with the fact, that every set has a number of > > > elements. > > > > Every set does have a specific number of elements. That's the > > cardinality of each set. > > > Yes For the finite case, yes. > > > > > > > > > You misinterpret totally when you say, I think there must be an > > > infinite natural number. I don't think so. I only argue that, if there > > > are infinite sets, there must be infinite natural numbers (since nat. > > > numbers are sets). > > > I don't say: there are infinite sets. You say: there are infinite sets > > > and there is no infinite number. And I say: If there are infinite sets > > > there must be infinite numbers. > > > > Now you seem to be saying that infinite natural numbers do exist. > > If you assume, there are infinite sets, in the sense that they are > actually complete, there must be a natural number which is infinite, > since natural numbers are sets. An actual complete set has an infinite > number of elements. But then you have a problem, since natural numbers > can't be infinite by definition. By what definition? Ain't the case. > So you have to expand the definition of natural numbers or you have to > deny the existence of infinte sets. > That's your problem, not mine. I choose the former. Why choose the latter? > > > > > > If you mean that infinite natural numbers exist, then you must prove > > this to be so. (Not infinite ordinals or infinite cardinals, since > > we know that these exist, but infinite naturals.) > > > > Or if you are saying that there are no such things as infinite sets, > > then you must prove this to be so. Prove that what everyone else > > thinks is an infinite set (such as N) does not really contain an > > infinite number of elements and is actually finite. > > > > Either way, I withdraw my apology. > > Either way, set theory with infinite sets is inconsistent. I disagree. What is inconsistent is the notion of cramming an infinite number of elements witha constant finite difference between each and its successor, into a finite value range. That is impossible. You cannot have an infinite set of unique finite whole numbers. > > > > > > > > My argumentation is very easy: > > > Every nat. number represents a set. If you look at the first 100 nat. > > > numbers, the 100th nat. number "100" represents the set {1, ... , 100}. > > > As this holds for every nat. number, > > > > This holds for all finite sets of finite natural numbers. > > This does not hold for infinite sets which have no largest member. > > So there is no actual complete infinte set. What about the reals in [0,1]. I guess you don;t consider them a set, even though there are individual real number elements within that range? What is the difference between an element of a set, and a point in space? Is a point not an element of space? Are there not an infinite number of points in any finite space? > > > > > > > > if there are infinite nat. numbers > > > there must be a infiniteth nat. number representing this set. > > > > That implies that there is a greatest member in the set that is an > > infinite natural. But this infinite set does not have a greatest > > member. So there is no reason for an infinite natural to exist. > > > > > > What's the reason of an actual complete set to exist? > > > > > But the definition of the nat. numbers with complete induction leads to > > > the consequence, that there could not be an infinite nat. number. > > > > > > That's the contradiction. > > > > Your assumption that the infinite set contains an infinite natural > > is what leads to your contradiction. Since this assumption is false, > > there is no actual contradiction. > > > > There is no assumption by me. > > > > > > So either the definition of nat. numbers must be changed or there is no > > > infinite set of natural numbers. > > > > Or your assumption must be changed. > > I've done no assumption. I use the assumptions of set theory and peano > axioms. Don't forget your diagonal proof!! ;) > > > > > > > > Or infinity must be interpreted in a completely other way. Not as a > > > size like you do. Infinity is just an unability to count it with > > > numbers because it runs out of all what we can know. > > > > "Infinity" has several meanings, and you're confusing at least two > > of them. An "infinite set" has a size represented by an infinite > > (transfinite) cardinal number. The limit of a series or sum that > > "approaches infinity" is a limit value that is larger than any real > > number. A "countably infinite" set is a set whose members can be > > enumerated (bijected) with the natural numbers. An "uncountably > > infinite" set is a set that has more members than the set of > > natural numbers. > > There is nothing more than endlessly, eternal, uncountable, infinite, > ... But the number of steps may be different, even over the infinite expanse. > > > > > > There is nothing illogical or inconsistent about saying that some > > sets contain more members than any finite number. > > > More than finite is infinite. But infinite is not a size. It's a state. > It's a word to depict the beginning of the unknown, the border of our > understanding. That's not a very rigorous notion, and I can't see it leading to mathematical results. > > > > > > > > All this is shown very expressive in my sketches at the start of this > > > thread. > > > > > > Why do you misinterpret all the time? Maybe my ability to express my > > > thoughts in english is too bad. > > > But why do you misinterpret Tony also? I think he is native english > > > speaker and you should be able to understand him. > > > > I don't think I'm misinterpreting what you're saying. I think what > > you're saying is wrong. What Tony is saying is certainly wrong. > > Set theory is inconsistent. > > > > > > > > > In this state there is no real problem with all this. aleph_0 is just > > > onother symbol for infinity. > > > > One particular kind of infinity, yes. > > > > > > > The problems occure in that moment if someone declares, that aleph_0 is > > > a size, which is greater than any nat. number. > > > > But that is the definition of Aleph_0; it is the size of the set of > > natural numbers (or any other set containing that amount of members). > > There is no amount "infinity". One can consider infinite N, when the properties of N are constant inductively. Where you have an inequality proven inductively, you must be sure the inequality does not shrink to a difference of 0 at N=oo. > > > > > > > > But there is no "greater" or "less than" or something like this. There > > > is just something other, something out of the things we could measure, > > > wigh or count. > > > The possibility of bijection don't say anything about the size of > > > infinity, > > > > The very fact that a bijection exists between set A and set B proves > > that the two sets are the same size. This is true for all finite and > > infinite sets. Bijection is the obvious way to show that sets are > > the same size. > > For the finite case: yes. In the infinite case, the mapping functions required must be used to compare the sets. > > > > > > > > since infinity is something sizeless, endless, countless. > > > That's all. > > > > Well, infinity is certainly a kind of "endlessness", yes. But it's > > not countless (if it's a countable infinity), and it's certainly > > not sizeless (since infinity is greater than any finite). > > Oh, it's just an accident of speaking to say: infinite many things are > uncountable many things? > You should have trust in the wisdom of language. > > Infinity is not greater than anything in any sense. It's just another > way to say: unnameable, nameless great. But it can be treated in a number of consistent ways. Unfortunately, transfinite cardinals and limit ordinals are not the way to go. > > Regards > AS > > -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/
From: Tony Orlow on 31 Oct 2005 13:43
albstorz(a)gmx.de said: > > Robert J. Kolker wrote: > > albstorz(a)gmx.de wrote: > > > > > > > > > > > I'm not shure if the reals build a set in spite of you and Cantor and > > > others are shure. > > > A set is defined by consisting of discrete, distinguishable, individual > > > elements. Now tell me: what separates a point on a line from the very > > > next point on the line to be discrete? > > > > There is no very next point under the ordinary ordering of reals. But > > given a pair of reals they are either equal or not. Between any two > > distinct real numbers there is always a third real different from the > > two given (with respect to the standard ordering of the reals). > > > > Bob Kolker > > Do you have a test on objects to proof if they are sets or not? Or is a > set just that what you want to have to be a set? > What's your definition of sets? Is the water in a can a set of water? > > Regards > AS > > It's a set of molecules, measured in moles. -- Smiles, Tony http://www.people.cornell.edu/pages/aeo6/WellOrder/ |