Cantor Confusion
in [Math]
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From: mueckenh on 13 Feb 2007 11:50 On 13 Feb., 13:11, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > >> >> > Given my fixed coordinate system, there remains no ambiguity at > >> >> > all. > > >> >> Then we are no longer talking about binary trees in general. > > >> > I never did so. > > >> I couldn't agree more. > > > Why then did you claim the opposite? Did you lie? > > Cause you insist using mathematically predefined words (trees, paths, > union) in a mathematical newsgroup. It is obviously true that you never > really never, really? > write about these mathematical issues. I write in the same sense about trees in that these trees exist. The only species of trees which I use does exist in any meaning of the word, i.e., really and mathematically. These trees contain nodes and edges and paths. These nodes and edges and paths can be enumerated by natural numbers in a way which supplies a unique means to determine the nodes of special paths and their unions. Your clumsy attempts to tangle up this clear proof are condemned to fail. Regards, WM
From: mueckenh on 13 Feb 2007 11:58 On 10 Feb., 15:31, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > If we decide to call the sparrow of E a number, then it > > > is not a natural number > > > But it does not mean that this sparrow is alive. > > If we decide to call a number between 1 and 2 a natural number, then > > this is a wrong definition. > > And if we decide to call the sparrow of E a natural number then > this is a wrong definition. So " If we decide to call the sparrow of > E a number, then it is not a natural number" Correct. And it can be shown that the sparrow is not lager than any natural number. Why should we decide to call it a number? Well, i is also called a number, but would we name it a number, if we had the choice today? > > > No statement you make about things that are true > > > of every set with finite cardinality, or things that > > > are true for every natural number, can be used > > > to show something about the sparrow of E. > > > The set E is not a set with finite cardinality > > > and the sparrow of E is not a natural number. > > > The fact that E is composed of sets with finite > > > cardinality does not mean that E is a set > > > with finite cardinality. > > > The finite cardinality has been proved by complete induction. > > No the finite cardinality of the components of E has been > proved by induction. E is not one of the components > of E, E is all components of E. What holds for all components (not only for each component!) holds for E. > and the fact that the compenents of E have a given > property does not mean that E has that property More precisely: If all initial segments of E have that property and if no element of E is outside of every initial segment, then E has that property. > > A limit of a sequence (a_n) has to have the Cauchy property. > > Piffle. We are not talking about the convergence of real numbers > to a real number. [Note also: it is the sequence that has the Cauchy > propery, not the limit] Here you are right. I meant the property |a_n - a| < eps but, in fact, Cauchy property is |a_n - a_m| < eps. > > omega - n > > = omega does not satisfy it. namely |n - omega| < eps > > No. One possible definition for the cardinality of E is the sparrow > of E. > The sparrow of E is a fixed equivalence class. Correct. It is the class of stes which have a natural number as cardinality which can grow. > So the statment > > [The cardinality of E] can only be the property that the natural > number > which is the cardinality of the set in present state can grow is > false. > > is false. Of course. Regards, WM
From: MoeBlee on 13 Feb 2007 12:58 On Feb 13, 2:49 am, mueck...(a)rz.fh-augsburg.de wrote: > On 12 Feb., 20:21, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > On Feb 10, 5:38 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 9 Feb., 21:26, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > > > > > Reality. > > > > > What reality? Empirically testable physical reality? A mathematical > > > > statement is not of that kind. > > > > Every mathematical statement is of that kind. > > > You snipped the rest of my argument regarding this. > > > > You would recognize this > > > (if you could, but you couldn't) if all reality ceased to exist. No > > > mathematics would remain. Yes, to grasp these facts, without such a > > > bad experience, i.e., during the continued existence of reality, one > > > needs some deeper thinking than is required to understand the meaning > > > of some crazy axioms. > > > Yes, if nothing existed, then mathematics would not exist, whatever it > > might be for there to be nothing that exists. Fortunately, we can do > > mathematics without consternation over such mind-boggling > > hypotheticals. > > You can do mathematics without seeing its foundation (which are not > axioms or logic but reality) but you cannot do mathematics without > using it. Whatever you mean by using reality but not seeing it, you have not shown any particulars regarding any significance you claim it has for proving theorems of mathematics. > Do you think it is wise to do something without being able or willing > to see its ground? Angels may be able to fly in the ether, > mathematicians are not. I don't require that mathematical statements themselves refer to physical objects. The mathematics developed can be USED for investigation of physical phenomneon, but the mathematical statements THEMSELVES are not statements about physical phenomenon (unless you can find physical objects that are themselves such mathematical objects). And I don't know of any bridges collapsing due to the set theory that axiomatizes the calculus used for building bridges. I am not arguing that one cannot do such calculus without an axiomatization; only that to the extent that one has an intellectual interest in knowing from precisely which axioms the theorems are proven, then set theory provides a recursive axiomatization. That is a primary part of my interest in mathematics; if it is not part of your interest, then so be it, but then you should recognize that none of your philosophizing does anything to satisfy my intellectual interest in axiomatic mathematics. MoeBlee
From: Virgil on 13 Feb 2007 15:17 In article <1171364856.226197.135140(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 12 Feb., 21:22, Virgil <vir...(a)comcast.net> wrote: > > In article <1171283208.696664.25...(a)a75g2000cwd.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 12 Feb., 04:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > In article <1171205918.124082.214...(a)a75g2000cwd.googlegroups.com> > > > > mueck...(a)rz.fh-augsburg.de writes: > > > > > > > On 11 Feb., 03:06, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > ... > > > > Well, I would concede that the above three things are representaions of > > > > the > > > > number three, using some convention. Anyhow, they are *not* the number > > > > three. > > > > > What has "the number three" that is not expressed above? > > > > Universality. Just as no single man represents mankind, > > I did not say that III expresses all numbers. It expresses all that > the number 3 can express. Not to me. Nor, apparently, to anyone except WM himself. So that what WM claims is universal is only so in a very small universe of his own. > > > no single > > instance of a set with three objects represents all sets of three > > objects, at least without common consent. > > What about all exsting sets with 3 objects, i.e., the fundamenal set > of 3? What fundamental set of 3 does WM refer to? In many set theories there cannot even exist such a "fundamental" set, and in such naive set theories as allow them there are worse problems than any WM has been able to find in ZFC or NBG.
From: G. Frege on 13 Feb 2007 15:23
On Tue, 13 Feb 2007 13:17:27 -0700, Virgil <virgil(a)comcast.net> wrote: >> >> What about all exsting sets with 3 objects, i.e., the fundamenal set >> of 3? >> > What fundamental set of 3 does WM refer to? > > In many set theories there cannot even exist such a "fundamental" set... > Yes. Neither in ZFC nor in NBG (and/or MK) there is such a set. But the approach works in NF and/or NFU. F. -- E-mail: info<at>simple-line<dot>de |