Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 18 Dec 2006 14:35 In article <1a591$458667c1$82a1e228$22650(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Virgil wrote: > > > In article <1166090594.020341.42340(a)80g2000cwy.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >>I do not understand why you argue that in > >>lim{x -> oo} lim{y -> oo} (2x + 3y)/xy = 0 = lim{y -> oo} lim{x -> oo} > >>(2x + 3y)/xy > >>interchanging limits is not possible. > > > > Nor do I. > > > > But for lim{x -> oo} lim{y -> oo} (2x + 3y)/(x + y) = 3 > > and lim{y -> oo} lim{x -> oo} (2x + 3y)/(x + y) = 2, > > one cannot exchange the order of the limits without changing the value > > of the result. > > This is highly misleading. What is misleading about a true and relevant statement? Both double limits exist but they have different values. The issue is whether such double limits are always reversible, and the answer, as demonstrated by the example above, is "NO".
From: Virgil on 18 Dec 2006 14:46 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. > 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. > 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. > 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. A element (E line (element in line)) but not E line (A element (element in line)) WM is pushing his quantifier dyslexia in our faces again.
From: Virgil on 18 Dec 2006 14:50 In article <1166449923.576810.297440(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1166359875.644197.31300(a)n67g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The difference between today's math and tomorrows math will be that the > > > physical and physiological foundations will be explored and taken into > > > account. > > > > WM would, if he had the power, allow only some of the words and none of > > the music. > > I would help to recognize all words which can be spelled out and to > distiguish them from such which never can be said. It would purify > present mathematics from superstition. WM's superstitious belief that "Ax Ey P(x,y)" and "Ey Ax P(x,y)" are necessarily logically equivalent, cripples his thoughts.
From: Virgil on 18 Dec 2006 14:57 In article <1166450382.369980.77190(a)t46g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > > > > Each line differs by 1 element from the preceding line. If the > > > > > > > number > > > > > > > of lines is actually infinite then the number of differences must > > > > > > > be > > > > > > > actually infinite too. > > > > > > > > > > > > So far so good. > > > > > > > > > > > Their sum is an infinite number. If all elements are there, then also > > > > > all sums are there. > > > > > > All of the /finite/ sums of an infinite series are always there , but > > the infinite sums are only "there" when the series converges. > > That does not prevent an infinite series from having infinitely many > > finite terms and infinitely many finite partial sums, even when it does > > not converge. > > 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". > 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".
From: Virgil on 18 Dec 2006 15:07
In article <1166451193.367851.138110(a)t46g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1166361248.951652.84540(a)l12g2000cwl.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > If you believe in a diagonal longer than any line of the matrix and in > > > paths which have no beginning, then I can't show you anything. > > > > You have things backwards, as usual. The reason that you cannot show me > > things is that you have not presented any axiom system on which to base > > them. When you want to prove something, you merely pull assumptions out > > of the air to serve as justifications. > > > > You cannot show me things within in ZFC or NBG that require you to > > assume things that contradict the XFC or NBG axioms because in an axiom > > system you are not allowed to make assumptions that contradict the > > axioms already assumed. > > If you are forced to believe in a diagonal longer than any line of the > matrix and in paths of the tree which have no beginning, then I can't > show you anything. While in ZFC and NBG, WM's "triangle" has only finite lines but an infinite diagonal, WM lies if he claims anyone has said that any path in a binary tree does not start at the root node. 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, and much that he claimed about mathematics that I find false. > > Anyhow, it is useless to discuss this topic further. I only wanted to > point out the fact that you are forced to believe these things. That > shows all I wanted to show. I choose to believe that certain things are true in ZFC+FOL because I have proved them in ZFC+FOL, or at least have seen the proofs of others in ZFC+FOL and found them valid. |