Cantor Confusion
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From: mueckenh on 19 Dec 2006 06:26 William Hughes schrieb: > > > > > > Element n+1 can be shown to exist in L_D (which is obviously a line > > > > > > containing n+1). > > > > > > > > > > No. L_D is bounded. The largest element of L_D is n. > > > > > L_D does not contain n+1. > > > > > > > > You misinterpret L_D. L_D is that line which contains all numbers > > > > contained in the diagonal. If your L_D does not contain them, then you > > > > have the wrong L_D. > > > > > > Assume that there exists an L_D which contains all the numbers > > > contained in the diagonal. L_D is bounded, > > > > If an unbounded diagonal exists, then obviously an unbounded line must > > exist. > > The conclusion is false, so the antecedent cannot be true. > > The diagonal is the potentially infinite set of natural numbers. > This exists and is unbounded. Yes, but not in the way you understand "to exist". Potentially infinite means: always finite. The diagonal is always finiter though not bounded. If it existed in the way you understand by "to exist", then also a line had to be infinite (by the definition: in the Equilateral Infinite Triangle, the diagonal consists of the ends of lines a_nn. A line has elements a_kn with k =< n and n eps N --- ALL n eps N), which is obviously wrong. > The antecedent is true therefore > the conclusion is true. But the object is not the diagonal. (Or your natural numbers are not a linearly ordered set.) Regards, WM
From: mueckenh on 19 Dec 2006 06:27 William Hughes schrieb: > Albrecht wrote: > > but in the same time |N isn't complete > > since |N is bijectable to the diagonal and the diagonal is never > > complete since there is no complete line (number) which covers the > > diagonal. > > No, the claim is that there is no line that covers the diagonal. Which results in the conclusion that there is no diagonal. Regards, WM
From: mueckenh on 19 Dec 2006 06:33 Virgil schrieb: > In article <1166449661.044809.225940(a)73g2000cwn.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > All elements that can be shown to exist in the diagonal can be shown to > > > > exist in one single line. > > > > > > Which line is that? > > > > That one which guarantees that a diagonal can contain all the elements > > which it contains. > > Why does such a supposed line need to exist? In ZFC no such line can > exist. > Because each element of the diagonal beongs to a line which contains all peceding elements. > > > It is wrong to assume that this could be guaranteed > > by many lines. Either it is guaranteed be one line or it is not the > > case at all. > > > > > That presumes a last line, which presumption is > > > unwarranted in ZFC and NBG and most other set theories. > > > > It presumes that all elements of the diagonal exist in the EIT. > > But WM's EIT presumes a last line, which is contrary to fact in ZFC. There are no "facts" in ZFC. > > > > > ZFC and > > NBG and most other set theories which wish to make us believe that all > > elements of the diagonal do exist although not in one line but > > distributed over many lines, are obviously wrong. > > They are only wrong if one imposes an assumption which makes them wrong. > > Absent such an assumption, there is noting to prevent them from being > right. > > > > > For linear sets it is > > impossible that an element exists in line m but not in line n > m. > > But quite possible in some line k with k > n > m. whch thebn can be called line m. > > > Threfore all elements which exists in smaller lines exist in a larger > > line. > > That every element is in some line does not require that some line > contain every element. In a linearly ordered set with only finite elements this is required. Otherwise give me a counterexampe. Regards, WM
From: mueckenh on 19 Dec 2006 06:40 Virgil schrieb: > > So, where is the sum 1+1+1+...+1 for infinitely many ones? > > If, as indicated by "1+1+1+...+1", there is a last "1" then there are > only finitely many of them, and the sum is well-defined. > Otherwise, such a sum is not defined or defineable. > > > > Where does the equation 1+1+1+...+1 = n cease to hold? > > When it ceases to be of form "1+1+1+...+1". I completely agree with you. Because then we have no longer finitely many differences. And then we have no longer a finite number n. Both statements are equivalent: Finitely many finite numbers and not finitely many not finie numbers. > > > There can be > > infinitely many finite right sides but not infinitely many finite left > > sides? Miraculous mathematics. > > Who said that? If WM says it, it is WM asking for miracles, but > mathematics does not ask it or claim it. > > Math, at least in ZFC or NBG, says there are infinitely many finite > values for each side of "1+1+1+...+1 = n". Precisely that's a good reason why this insane theory should be abolished. Infinitely many ones added together yield an infinite number, not a finite number as you wish to make me believe. Regards, WM
From: mueckenh on 19 Dec 2006 06:42
Virgil schrieb: > And WM is right that he cannot show me anything, as there is nothing > true about mathematics that he has so far claimed that I do not already > know, What is your present opinion about the ordinal 2^omega? Regards, WM |