Cantor Confusion
in [Math]
|
From: Virgil on 11 May 2007 16:07 In article <1178872656.637980.112800(a)n59g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 21:30, Virgil <vir...(a)comcast.net> wrote: > > In article <1178796129.155631.279...(a)u30g2000hsc.googlegroups.com>, > > > > > Yes. But why only see the one side of the medal? For every path p > > > > > there is another path *existing in the tree* which is not different > > > > > from p at any node, because p is not single at any node. > > > > What the above says is that for every path p there is a path p' in the > > same tree which is the same as p at every node but somehow not as a > > whole the same as p. > > > > But that is false in ZF or NBG trees. > > > > Whatever have I said that WM claims would be wrong > > The line preceding your question is wrong. Then is WM claiming that "for every path p there is a path p' in the same tree which is the same as p at every node but somehow not as a whole the same as p." is NOT false, at least in ZF ands NBG? > There are not more nodes than we can ask for. WE can ask for all infinitely many.
From: Virgil on 11 May 2007 16:13 In article <1178873334.517622.92460(a)e51g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 21:36, Virgil <vir...(a)comcast.net> wrote: > > In article <1178806498.433405.226...(a)o5g2000hsb.googlegroups.com>, > > > > > > > The axioms of ZF tell how to specify sets in ZF. > > > > > Yes, how to specify, but they do not specify sets except few. > > > > And having those few, with intersections, differences, etc., one can > > build many without a single "rule" of the sort WM implies are necessary. > > Just the description of how you apply these axioms to "build" a set, > precisely _that is the rule_ or formula. The rules or formulas, as implied by the "rule, domain, range" supposed definitions describe how to determine a "value" in the range for a given "argument' in the domain. So unions, intersections, etc, of sets of ordered pairs don't count. > > Show me the forest. There are too may trees. I can't see it. A fairly accurate self analysis by WM.
From: Virgil on 11 May 2007 16:16 In article <1178873827.224574.159920(a)w5g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 23:03, William Hughes <wpihug...(a)hotmail.com> wrote: > > > Note: you can enumerate an infinite set of nodes. > > Let us assume that. > > > > For every node n belonging to the set of nodes that you can enumerate, > > one path p'(n) is sufficient to accompany p up to that node. > > p'(n) is not the same for every node. > > For which node is another path required? The first one (at the lowest level) in p' but not in p. And if there is n such first one, then p = p'.
From: Virgil on 11 May 2007 16:17 In article <1178873959.166862.5470(a)q75g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Mai, 00:07, Virgil <vir...(a)comcast.net> wrote: > > > Would you call mathematics an art? > > > > Yes! > > And what about theology? Is it a science or an art or something else? That seems to be more WM's specialty than ours.
From: Virgil on 11 May 2007 16:22
In article <1178874428.056116.33930(a)e65g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 11 Mai, 00:22, Virgil <vir...(a)comcast.net> wrote: > > In article <1178830047.112233.306...(a)p77g2000hsh.googlegroups.com>, > > > > > > > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 10 Mai, 21:36, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1178806498.433405.226...(a)o5g2000hsb.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 10 Mai, 03:59, Virgil <vir...(a)comcast.net> wrote: > > > > > It is irrelevant what it is according to your opinion or "in the first > > > place". > > > > English definitions of function, at least those in the more rigorous of > > texts, insist that they be sets of ordered pairs, and "single-valued" in > > the sense that if <a,b> and <a,c> are members then b = c, but do not > > insist on there being any rule by which they are defined. > > If a set of ordered pairs is defined, there must be a formula or > whatever you may call it, defining it. Otherwise it is undefined. only if one defines every such definition as a formula. > > Once defined, one can then define other related sets, such as a domain, > > codomain or range, but these are only consequences of the original > > definition, as are such properties as being injective or surjective. > > > > I said that a function consists of formula, domain and range. And mathematics formally defines functions otherwise, for historical reasons having to do with the flaws in defining functions your way. Live with it. |