Cantor Confusion
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From: stephen on 21 Oct 2006 01:51 Mike Kelly <mk4284(a)bris.ac.uk> wrote: > Han de Bruijn wrote: >> Dik T. Winter wrote: >> >> > I think that, compared to Cantor, in modern set theory potential and actual >> > infinity are split up again. The contents of the set N form only a potential >> > infinity, on the other hand, the *size* is an actual infinity. >> >> And you call _that_ "thinking" ?! >> >> Han de Bruijn > Hey look, Han is resorting to insults. What a hypocrite! > -- > mike. Are you surprised? Stephen
From: mueckenh on 21 Oct 2006 06:44 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > "Continue in this manner" is just what is used to describe how the > > infinite is realized. > > Not in formal axiomatic set theory. If you can't define your "continue > in this manner" in the language of set theory, then your argument is > not, contrary to your claim, in set theory, and so, if your argument > does depend on "continue in this manner" and you won't define it in > some set theory such as Z set theory, then, of course, we're done as > far as me taking you seriously in regards to your argument and claim > that it shows anything about set theory. "Continue in this manner" is Cantor's definition, namely: "... und es erfährt daher der aus unsrer Regel resultierende Zuordnungsprozeß keinen Stillstand." My translation: The process does not come to halt (or does not stand still). Cantor laid the foundations of sets theory. If modern set theory would not accept his definition, then it should not call itself set theory. But this same definition is the *only* one defined in ZFC: There exists a set such that x u {x} is in it if x is in it, abbreviated according to Peano: if n is in it, then n+1 is in it. No other than this "continue in this manner"-definition can be claimed correct in modern set theory. If you claim that what you call modern set theory has a deviating definition of infinity, then I am not interested in your theory. You (not I) may translate my arguing into your theory, if you cannot understand other languages. if your translation leads to my results, you may be satisfied, and if it does not, then your translation or your theory is a mess. =========================== > You don't know what your're talking about. The convergence is proven. The identity of 1 and 0.999... in analysis is proven. The difference 10^(-n) disappears for n --> oo (and not earlier!). The identity is not proven in case of the diagonal number because it cannot be proven, because of the lacking factor 10^(-n). Even for the digit with index n --> oo the difference is as important as for the fist digit. Here is the schisma, the point where mathematics is inconsistent: Either the digit becomes negligible in both cases or in none of the cases, i.e., either 1 =/= 0.999... in analysis or the later digits of the diaonal number become more and more unimportant such that for an infinite number there are negligible digits and Cantor's argument breaks down. Regards, WM
From: mueckenh on 21 Oct 2006 06:47 William Hughes schrieb: > > > No. If you restrict yourself to computable functions you have some > > > counterintuitive results. Assume that > > > the language you are working in has a finite alphabet. Then the set > > > of all finite strings in the language is listable using a computable > > > function > > > (use dictionary order). And so the set of all finite strings is > > > countable. > > > Now, A, the set of all strings which define a computable number is a > > > subset of the set of all finite strings. So A is countable, right? > > > Wrong! > > > It is not true that every subset of a countable subset is countable. > > > > It is true that every set can be well ordered and that any two sets can > > be compared. Both are equivalent or one is equivalent to a sequence > > (ordered subset) of the other. What you say is not counter intuitive > > but wrong. No response? Here we work in our standard model, not in some externally finite but internally uncountable model. > > > > > > And that set can't be listed. > > > > > > > > > > And here is your problem. Uncountable means unlistable. > > > > > > > > Not my problem. Countably infinite means unlistable too. > > > > > > Yes, but what does unlistable mean? > > > > Listable means countable. > > A tad circular? An identity. > A set B is is listable (and therfore countable) > if there exists a function, f, > from the natural numbers to the set B. Whether we can immediately > conclude that a subset of B is listable (and therefore countable) > depends on what types of functions f we allow. Why should we refuse our admission for some functions? Well, we must restrict our allowance in some cases to satisfy Skolems requirement that every consistent theory has a finite model. I never heard of more ridiculous nonsense than this "explanation" of missing functions to guarantee internal uncountability of an externally finite model. I am not willing to waste my time by considering such a mess. Regards, WM
From: Sebastian Holzmann on 21 Oct 2006 06:51 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > I am not willing to waste my time by considering such a mess. Please don't. Your future career in basket weaving will be much more fun for both you and us.
From: mueckenh on 21 Oct 2006 08:47
Virgil schrieb: > In article <1161378001.475899.279610(a)f16g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Every diagonal number is constructed and, therefore, is constructible. > > > > > > When constructed from a constructable list. > > > > What is an unconstructable list? Do you call any undefinite mess a > > list? > > Any list is construtced. Any diagonal number is constructed. > > A list of reals for the Cantor construction need not be made up of > constructable numbers, it is only required that the nth number be > constructable to the nth decimal place. It is assumed that nearly all numbers of the list are contructible to more than the n-th digit. But the main point is: It is assumed that the diagonal number is a constructed number. The set of constructed numbers, however, is countable. Regards, WM |