Cantor Confusion
in [Math]
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From: William Hughes on 7 May 2007 15:32 On May 7, 3:19 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > That is not the point here. The point is that there are subsets of > countable sets which do not allow for a bijection with N. No. there are subsets of a finitely definable set which are not finitely definitable. The difference is... M: WAHH! WAHH! WAHH! I'm not listening. - William Hughes
From: Virgil on 7 May 2007 15:36 In article <5hJ%h.160241$kJ4.150295(a)reader1.news.saunalahti.fi>, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: > On 2007-05-07, in sci.math, Virgil wrote: > > Some definitions of pi are infinite but quite well understood. > > Really? What definitions would these be? The ones that define it as the limit of an infinite sequence, as such processes are, in theory, infinite, but usually in practice soon close enough for all practical purposes. While the process can, in some senses, be "finitely defined", the exact value cannot be finitely determined.
From: WM on 7 May 2007 15:38 On 7 Mai, 18:53, Virgil <vir...(a)comcast.net> wrote: > In article <1178537072.163162.176...(a)u30g2000hsc.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 7 Mai, 00:59, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > Look! Over there! A pink elephant! > > > That seems necessary to make us believe that a paths that shares > > every node with another path does not share every node with another > > path. But it is not sufficient. > > > Regards, WM > > If WM believes that one path in any tree can share every node with > another and different path, he has passed beyond reason into some sort > of fugue state. Can you explain, how every node of p is shared by other paths, but not every node is shared by at least one path p'? Do the companion paths alternate? E.g., p' for every even node and p'' for every odd node? Regards, WM
From: WM on 7 May 2007 15:42 On 7 Mai, 18:50, Virgil <vir...(a)comcast.net> wrote: > In article <1178536779.509069.124...(a)l77g2000hsb.googlegroups.com>, > There are all sorts of things in naive set theory texts that formal set > theories do not allow. I am not interested in formal set theory but only in the question: Are there uncountably many real numbers? > > Fn = (2-1-1) > > It is the number of separated path coming out of the n-th element of > > the tree minus the number of ingoing separated paths minus the number > > of nodes of this element. > > In that case, it is irrelevant. Why? Regards, WM
From: WM on 7 May 2007 15:45
On 7 Mai, 18:50, Virgil <vir...(a)comcast.net> wrote: > In article <1178536779.509069.124...(a)l77g2000hsb.googlegroups.com>, > > > 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. > > > It is a definition from a book on set theory, called "Introduction to > > Set Theory". So it gives basic set theory. > > It may very well be a very naive set theory for non-mathematicians. > Who wrote it, and for what sort of students is it supposed to be an > introduction? Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" Marcel Dekker Inc., New York, 1984, 2nd edition. 250 pages. For students of set theory. Regards, WM |