Cantor Confusion
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From: mueckenh on 20 Oct 2006 16:42 William Hughes schrieb: > mueck...(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > A constructible number is a number which can be constructed. Definition > > > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > > > the proof, as if the method were purely existential, i.e. as if the > > > > proof, while showing that there exist decimals belonging to C but not > > > > to C0, did not allow to construct such decimals." > > > > > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > > > or the diagonal of a list is that what I call constructible. If you > > > > dislike that name, you may call these numbers oomflyties. Anyhow that > > > > set is countable. > > > > > > Nope. By the definitions you use, that set is not countable. > > > > > Every set of constructions is countable due to the finite alphabet of > > any language. > > > 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. > It is not true that there is a computable function which will list the > elements of A (to do this you would have to be able to identify the > elements > of A, and to do this you have to solve the halting problem). > > > > > > > And that set cannt 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. In set theory listable and countable is obviously the same as unlistable and uncountable. Regards, WM
From: mueckenh on 20 Oct 2006 16:46 MoeBlee schrieb: > > But the understanding of them is a prerequisite of understanding > > infinite sets (according to Cantor who new more about that matter en > > Zermelo, Fraenkel and v. Neuman together. Therefore it is no wonder > > that set theorists easily intermingle both). > > Understanding of them is a philosophical concern, and such > philosophical inquiries are not precluded by set theory; only that the > philosophical investigations of differences between actual and > potential infinity do not occur WITHIN set theory; This shows that set theory is completely useless, because the only topic to be dealt with successfully is infinity. > > > MB: > > I just asked a questions. > > > > More basically, what is your context? Is your argument about binary > > trees supposed to be within set theory or not? > > > > WM: > > As far as I know all mathematics including the use of fractions is > > within set theory or derived from it. > > Okay, so I take it that your argument about that binary tree is meant > to be within say, Z set theory. Yes, I do wish to be educated by you. > Rather than just giving an ostensive definition of the relation with > edges and paths "continuing in this manner to infinity", would you > please give a rigorous definition of the relation? "Continue in this manner" is just what is used to describe how the infinite is realized. Consider a finite path with n edges. This path carries a load of (1 - (1/2)^n)/(1 - 1/2) own edges. For n --> oo this yields a load of 2 edges. It is very simple, it derived from current mathematics, which is derived from your axioms, therefore it must not further be confused by retranslating it with your axioms. Please follow my discussion with jpale who understood the problem. > The diagonal argument does not contradict that 1=.999... No, it is even unaware of this fact, because the necessary convergence is missing. Regards, WM
From: mueckenh on 20 Oct 2006 16:48 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > I've never seen "potential infinity" or "actual infinity" in any > > > textbook I've used. > > > > So you read not the right books or too few. > > 'potentially infinite' and 'actually infinite' are mentioned often in > philosophy and history of mathematics. But would you please just refer > to a single textbook of set theory, analysis, or calculus that gives > mathematical definitions of 'potentially infinite' and 'actually > infinite'? The neglection of meaning is not an advantage of modern mathematics. In fact there are few modern books mentioning the difference. One is Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976) p. 6 "the statement lim 1/n = 0 asserts nothing about infinity (as the ominous sign oo seems to suggest) but is just an abbreviation for the sentence: 1/n can be made to approach zero as closely as desired by sufficiently increasing the integer n. In contrast herewith the set of all integers is infinite (infinitely comprehensive) in a sense which is "actual" (proper) and not "potential". (It would, however, be a fundamental mistake to deem this set infinite because the integers 1, 2, 3, ..., n, ... increase infinitely, or better, indefinitely.)" and later: "Thus the conquest of actual infinity may be considered an expansion of our scientific horizon no less revolutionary than the Copernican system or than the theory of relativity, or even of quantum and nuclear physics." Regards, WM
From: mueckenh on 20 Oct 2006 16:50 jpalecek(a)web.de schrieb: > mueckenh(a)rz.fh-augsburg.de napsal: > > jpalecek(a)web.de schrieb: > > > > > > The set of constructible numbers is countable. Any diagonal number is a > > > > constructed and hence constructible number. > > > > > > No. Definition (from MathWorld): Constructible number: A number which > > > can be represented by a finite number of additions, subtractions, > > > multiplications, divisions, and finite square root extractions of > > > integers. > > > > > > How do you represent the diagonal number (which is sort of a limit of a > > > series) > > > via FINITE number of +,-,*,/,sqrt( ) ? > > > > Which digit should not be constructible by a finite number of > > operations? > > No digit, but the number has to be constructed (according to MathWorld > definition). So you must do something like > > diag=x^2+sqrt(y+z/x) Either such a formula or a full description like the diagonal number of a list. The number of formulas and descriptions is countable. > > > > > > > > Every list of reals can be shown incomplete in exactly the same way as > > > > every list of contructible reals can be shown incomplete. > > > > > > No. > > > > A constructible number is a number which can be constructed. Definition > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > the proof, as if the method were purely existential, i.e. as if the > > proof, while showing that there exist decimals belonging to C but not > > to C0, did not allow to construct such decimals." > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > or the diagonal of a list is that what I call constructible. If you > > dislike that name, you may call these numbers oomflyties. Anyhow that > > set is countable. And that set cannt be listed. Therefore the diagonal > > proof shows that a set of countable numbers is uncountable. > > I like the name, but your "definition" isn't a definition. That depends on how you define "definition". For me it is a sufficient definition. If not for you, then translate the quote of Fraenkel and Levy as you like. If you do it without error, the result will be the same. Anyhow, a diagonal number is a number which can be or has been constructed. Regards, WM
From: William Hughes on 20 Oct 2006 16:51
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > A constructible number is a number which can be constructed. Definition > > > > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > > > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > > > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > > > > the proof, as if the method were purely existential, i.e. as if the > > > > > proof, while showing that there exist decimals belonging to C but not > > > > > to C0, did not allow to construct such decimals." > > > > > > > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > > > > or the diagonal of a list is that what I call constructible. If you > > > > > dislike that name, you may call these numbers oomflyties. Anyhow that > > > > > set is countable. > > > > > > > > Nope. By the definitions you use, that set is not countable. > > > > > > > Every set of constructions is countable due to the finite alphabet of > > > any language. > > > > > > 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. > > > It is not true that there is a computable function which will list the > > elements of A (to do this you would have to be able to identify the > > elements > > of A, and to do this you have to solve the halting problem). > > > > > > > > > > And that set cannt 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? 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. - William Hughes |