Cantor Confusion
in [Math]
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From: MoeBlee on 6 May 2007 20:31 On May 6, 9:05 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 6 Mai, 11:34, MoeBlee <jazzm...(a)hotmail.com> wrote: > What you say is insane. It's not insane just to point out that a proof in formal Z set theory is one that uses only first order logic applied to the axioms of formal Z set theory. Here's what I wrote, which you did not address: "(1) If a presentation of a proof uses any principles or premises whatsoever other than first order logic applied to the Z axioms, then that presentation is not that of a Z set theory proof. So, to reiterate, since you are so very obtuse, if you find that some presentation or another uses some premises or principles other than first order logic applied to the Z axioms, then that presentation is not of a Z set theory proof. (2) Usual textbook presentations of the theorem that no set maps onto its power set use nothing but first order logic applied to the axioms of Z set theory." There's nothing that is insane about any of that. Indeed, what I mentioned are simple matters that you need to understand. > > (3) In ordinary set theory, a function is a certain kind of set of > > ordered pairs and, in ordinary set theory, a function is NOT a triple > > of a formula, domain, and range. That you even think a set theoretic > > function must have a FORMULA (!) as a component shows, again, > > your > > lack of understanding even the basics of set theory. > > Function, as understood in mathematics, is a procedure, a rule, > assigning to any object a from the domain of the function a unique > object b, the value of the function at a. A function, therefore, > represents a special type of relation, a relation where every object a > from the domain is related to precisely one object in the range, > namely, to the value of the function at a. That is a very common NON-formal definition and notion of function found in a great amount of mathematics. However, it is NOT the set theoretic defintion that is being used in formal Z set theory and is NOT the definition that is used in formal Z set theory to prove Cantor's theorem that there is no function from a set onto its power set. If you INSIST on using 'function' in a sense that is NOT the sense in formal Z set theory, then you are not talking about formal Z set theory. > Please do not believe that I will repeat for another 1000 postings > your controversy with other great mathematicians about the properties > of a function. All you need to do is pick up a textbook on set theory to see that in ordinary set theory 'function' is defined to be a certain kind of set of ordered pairs and not as you define. > > (4) I don't know about Hessenberg's presentations, but in ordinary > > presentations, even though we don't need to prove by contradiction, it > > is sometimes easy to set the proof up that way. Then from the reductio > > assumption of a function mapping N onto PN, as to the set of > > elements > > of N that are not mapped to a member of PN, we PROVE that set > > exists > > by a simple use of the axiom schema of separation. That you even > > question that shows that you do not understand even such basic > > things > > about set theory as the axiom schema of separation. > > > (5) As to the function from N onto PN, again its existence is either a > > reductio assumption, or, by modus tollens or some other principle that > > is included in first order logic, we prove such a function does not > > exist. > > Best by avoiding any formula and any clear idea what you do. No, a function is not ITSELF a triple of a formula, domain, and range, but a formula may DEFINE a certain set of ordered pairs that is a function. You need to understand such basics of set theory. > > > You are wrong to assume that I did not study set theory. > > > I infer it by the ignorance you display. > > Obviously you are not very good in concluding from given facts on > hidden facts. I have no idea what "hidden facts" you're thinking of. > > > But the > > > formalism veils the problem it is built upon. > > > Whatever you think that means, my point stands: Proofs of theorems > of > > formal Z set theory make no use of anything other than first order > > logic applied to the axioms. > > Then use your axioms to understand the binary tree. I offered to consider your tree argument if you would only tell me from the outset certain things about your intended context and terms of discussion. You refused. I am not at all inclined to enter into a quagmire of argument in which you will not even agree as to defintions that we can trace back to primitives and axioms. But if you ever get around to stating your logisitc system, axioms, primitives and SYSTEMATICALLY given defintions, then please do let me know. > > You've not given any set theoretic definition of > > "2-1-1+2-1-1+2-1-1+-... " > > Everybody who has studied a bit of set theory should know > 2-1-1+2-1-1+2-1-1+-... < 2+1+1+2+1+1+2+1+1+... = alep_0 > So whatever misunderstanding you may want to apply, > 2-1-1+2-1-1+2-1-1+-... < 3 < aleph_0. I said you didn't define "2-1-1+2-1-1+2-1-1+-... " And you still haven't. > > If you want to talk about an infinite summation, then you need to be > > able to eliminate the '...' and put it in the form: Sum(from n to oo) > > F(n). > > Sum [n in N] (2-1-1) < 3. > That is but another way of writing 2-1-1+2-1-1+2-1-1+-... < 3. That is not a DEFINITION of "2-1-1+2-1-1+2-1-1+-... " > If you have problems with infinite sums or products then you should > look up set theory, for instance Koenigs theorem. I have no problem with infinte sums and products. I am just pointing out that your notation is NOT justified as infinite summation until you specify what F is in Sum(from n to oo) Fn. As I said, set theory is not answerable to your fingerpainting with symbols. Nor am I. MoeBlee
From: WM on 7 May 2007 06:16 On 6 Mai, 20:16, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 6, 10:53 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > Fine. That means, the set is countable but there is no bijection with > > N definable. > > No it means that while a bijection exists it is not finitely > definable. The bijection with this countable set is not finitely definable. Why then do you require a finitely definable bijection between paths and nodes in the tree? Regards, WM
From: WM on 7 May 2007 06:21 On 6 Mai, 22:10, Virgil <vir...(a)comcast.net> wrote: > > > > > > Why then do you believe that Cantor's diagonal proof is true? > > > > > > You'd have to define 'true proof'. Meanwhile, with just some basic > > > > > knowledge of predicate calculus and set theory, one can see that > > > > > Cantor's diagonal argument is formalizable into a proof in first order > > > > > logic from the axioms of Z set theory. > > > > > There is an implicit assumption entering which is wrong. > > > > The difference of 1 between the digits of two numbers is insufficient > > > > to distinguish these numbers in the limit n --> oo. > > > > What WM is describing is not a difference between two fixed numbers but > > > between two sequences of numbers whose differences converge to 0, the > > > limiting difference is indeed zero but that is totally irrelevant, > > > not for the sequence of digts of the diagonal number and any other > > entry of the list. > > Since the "diagonal" differs from each listed member at some finite > place value, what happens at later place values is irrelevant to whether > they are equal or different. The number of separated paths is les than the number of nodes at each finite level. What happen after each finite level is irrelevant. > > While there may be some subsequence of the original which converges to > the diagonal, no member of that subsequence EQUALS that diagonal, which > is all that is required. That is not different in case of 0.999... and 1.000... Why must it be excluded? > > > > > > In base b, numbers of form m/b^n have two expansions. But excluding > > > these, a difference of 1 or more in any digit position differentiates > > > between two different reals. > > > It does only differentiate the reals at a finite index. But at a > > finite index it differentiates as well between 0.999... and 1.000.... > > So, why do you want to exclude these but not those? > > The rule by which the diagonal is constructed makes it differ from both > representations when those representations have equal values. The rule does not make 1.000... differ from 0.999.... > > In any case, for the set of all binary sequences (not numbers) no two > are the same, so the issue does not arise I am interested in the number of real numbers only. Regards, WM
From: WM on 7 May 2007 06:23 On 6 Mai, 22:26, Virgil <vir...(a)comcast.net> wrote: > In article <1178461785.380972.277...(a)e65g2000hsc.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 6 Mai, 03:53, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1178190762.081101.159...(a)h2g2000hsg.googlegroups.com> WM > > > <mueck...(a)rz.fh-augsburg.de> writes: > > > > Wrong. If there is always a path p' with p, > > > > What does this statement mean? > > > Even in the *infinite* tree a path cannot be distinguished from all > > other paths. That means, a real number which cannot be described by a > > finite formula (and most of them cannot) does not exist. > > What axiom requires that a number must be defined by a finite formula in > order to exist? > What kind of existence has an undefinable number? > > > Cantor's diagonal proof fails, because the diagonal number is never > > distinguished from all other real numbers (if uncountably many real > > numbers exist). > > Cantor's "diagonal" proof only involves binary strings, not numbers, and > a very simple rule distinguishes the "diagonal" form every member of the > list. Only finite strings have been tested so far. Regards, WM
From: WM on 7 May 2007 06:43
On 6 Mai, 22:33, Virgil <vir...(a)comcast.net> wrote: > In article <1178463181.745792.46...(a)y80g2000hsf.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 6 Mai, 04:40, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1178366406.361131.321...(a)y5g2000hsa.googlegroups.com> WM > > > <mueck...(a)rz.fh-augsburg.de> writes: > > > ... > > > > Theorem 2: The set of finitely defined numbers cannot be put into a > > > > list. > > > > > Proof: Assume such a list exists. This means, such a list has been > > > > finitely defined. > > > > Wrong. Existence does *not* mean finitely definable. > > > Marthematical existence does mean definable. > > But not necesssarily "finitely" so. Who can give or understand an infinite definition? > > > What else should it mean? > > Definitions can only be finite. > > That may be one of WM's axioms, but does not appear in ZF or NBD. That is the basis of logic which ZF or NBG have to obey (if they are logical systems). Regards, WM |