Cantor Confusion
in [Math]
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From: WM on 10 May 2007 16:47 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: > > > In article <1178738274.909825.208...(a)e65g2000hsc.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 9 Mai, 20:16, Virgil <vir...(a)comcast.net> wrote: > > > > > In article <1178707943.860551.269...(a)h2g2000hsg.googlegroups.com>, > > > > > > > But the formal definition does not specify how a and b are related > > > > > > other than by mentioning F. This F however is undefined unless you > > > > > > know from the first paragraph that it is a procedure or rule. > > > > > > If it is a set, it is defined as sets are defined, which does not > > > > > require anything as undefined as "procedure" or "rule". > > > > > How do you define a set, unless you cannot specify the elements > > > > belonging to it? > > > > I am reasonably certain that that question does not mean in standard > > > English, what you intended to ask. > > > Should have been: > > How do you define a set, unless you can specify the elements belonging > > to it? > > > > 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" f the sort WM implies are necessary. > > > > > > > > > > > > > Further definition 3.1 contains: It is customary to use the notations > > > > > > / > > > > > > F: A -> B/. Why do you think A and B were not two sets and F was not > > > > > > a > > > > > > formula > > > > > > Define "formula". Until it is defined, it has no mathematical > > > > > significance, and it hasn't yet been defined. > > > > > A formula is a rule or prescription specifying or determining which > > > > element of the domain is mapped on which element of the range. The > > > > formula in its widest sense can even consist of several expressions. > > > > In case every element of the domain has to be mapped by its own > > > > expression, the formula can even be a catalogue or set. > > > > So a set can be defined by being a set! How marvelously circular! > > > A set can be specified by specifying its elements. This can be done > > very well, in particular for finite sets, by a list. Yes, the rule or > > formula can be a list listing the elements of a set. > > > > > Do you know what a relation is? Do you know that there is also the > > > > domain and the range required? > > > > There are a domain and range to a relation, but they are consequences of > > > its definition, not antecedents to it. > > > Answer only by yes or no > > The questions were irrelevant to the issue of whether a function is, > first of all, a set of ordered pairs.- Zitierten Text ausblenden - It is irrelevant what it is according to your opinion or "in the first place". I said that a function consists of formula, domain and range. This is true and provable, whatever you try to counter argue. It is taught in mathematics in Germany at every university and of course it is also taught by me at the university of applied sciences Augsburg. If you would try to study mathematics in Germany and would not know that a function *necessarily* consists of domain and range (in German Definitionsbereich and Wertebereich, which clearly shows that a function without domain is not even defined !), you could hardly obtain a Bachelor degree. Regards, WM
From: MoeBlee on 10 May 2007 16:55 On May 10, 1:47 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > I said that a function consists of formula, domain and range. > This is true and provable, whatever you try to counter argue. Provable? In what system of logic and from what axioms? Define, in set theory, 'consists of'. MoeBlee
From: William Hughes on 10 May 2007 17:03 On May 10, 4:06 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: <snip> > > Finity or infinity does not play a role. The only relevant > property is: "for all nodes which you can enumerate". > So one can enumerate an infinite set of nodes. (otherewise finity or infinity would have to play a role) Note: you can enumerate an infinite set of nodes. 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. Look! Over there! A pink elephant! For all nodes you can enumerate, one path p' is sufficient to accompany p. - William Hughes
From: Virgil on 10 May 2007 18:07 In article <1178829465.990283.182080(a)q75g2000hsh.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 04:15, Virgil <vir...(a)comcast.net> wrote: > > In article <1178739230.073383.98...(a)o5g2000hsb.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 9 Mai, 20:25, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1178716861.173911.234...(a)y5g2000hsa.googlegroups.com>, > > > > > > WM has no power to constrain mathematics to what he thinks it should be. > > > > > > That it is not what WM thinks it should be is by now apparent, but that > > > > it is quite productive anyway is equally apparent > > > > > Producing what? Yes, the working mathematicians, they are quite > > > productive, but they do not need transfinite set theory. Transfinite > > > set theory is the most useless science ever created. > > > > That set theory is mathematics, and therefore not science at all, seems > > to have escaped WM. > > In Germany mathematics belongs to "Wissenschaft" which is usually > translated by science. At the university where I studied (and at most > others in Germany), there is a common faculty of mathematics and > natural sciences. One hundred years ago this was a world centre of > mathematics and physics: Hilbert, Klein, Minkowski, Zermelo, Weyl, > Born, Heisenberg, Pauli worked there. Several years earlier, also > Gauss, Riemann and Dedekind. Would you call mathematics an art? Yes! In English, "science" has come to mean the "natural" sciences, those whose "truth: is judged by appealing to the nature of the universe, and excluding at least "pure" mathematics and at most ambiguous about applied mathematics.
From: Virgil on 10 May 2007 18:22
In article <1178830047.112233.306950(a)p77g2000hsh.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>, > > > > > > > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 10 Mai, 03:59, Virgil <vir...(a)comcast.net> wrote: > > > > In article <1178738274.909825.208...(a)e65g2000hsc.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 9 Mai, 20:16, Virgil <vir...(a)comcast.net> wrote: > > > > > > In article <1178707943.860551.269...(a)h2g2000hsg.googlegroups.com>, > > > > > > > > > But the formal definition does not specify how a and b are > > > > > > > related > > > > > > > other than by mentioning F. This F however is undefined unless > > > > > > > you > > > > > > > know from the first paragraph that it is a procedure or rule. > > > > > > > > If it is a set, it is defined as sets are defined, which does not > > > > > > require anything as undefined as "procedure" or "rule". > > > > > > > How do you define a set, unless you cannot specify the elements > > > > > belonging to it? > > > > > > I am reasonably certain that that question does not mean in standard > > > > English, what you intended to ask. > > > > > Should have been: > > > How do you define a set, unless you can specify the elements belonging > > > to it? > > > > > > 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" f the sort WM implies are necessary. > > > > > > > > > > > > > > > > > > > > > Further definition 3.1 contains: It is customary to use the > > > > > > > notations > > > > > > > / > > > > > > > F: A -> B/. Why do you think A and B were not two sets and F was > > > > > > > not > > > > > > > a > > > > > > > formula > > > > > > > > Define "formula". Until it is defined, it has no mathematical > > > > > > significance, and it hasn't yet been defined. > > > > > > > A formula is a rule or prescription specifying or determining which > > > > > element of the domain is mapped on which element of the range. The > > > > > formula in its widest sense can even consist of several expressions. > > > > > In case every element of the domain has to be mapped by its own > > > > > expression, the formula can even be a catalogue or set. > > > > > > So a set can be defined by being a set! How marvelously circular! > > > > > A set can be specified by specifying its elements. This can be done > > > very well, in particular for finite sets, by a list. Yes, the rule or > > > formula can be a list listing the elements of a set. > > > > > > > Do you know what a relation is? Do you know that there is also the > > > > > domain and the range required? > > > > > > There are a domain and range to a relation, but they are consequences > > > > of > > > > its definition, not antecedents to it. > > > > > Answer only by yes or no > > > > The questions were irrelevant to the issue of whether a function is, > > first of all, a set of ordered pairs.- Zitierten Text ausblenden - > > 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. 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. There need not be a 'formula', in any formal sense, in evidence in order to have a function, but one does need a set of ordered pairs. > This is true and provable It is only provable when it is assumed. > If you would try to study mathematics in Germany and would not know > that a function *necessarily* consists of domain and range (in German > Definitionsbereich and Wertebereich, which clearly shows that a > function without domain is not even defined !), you could hardly > obtain a Bachelor degree. From you I would not accept one. |