Cantor Confusion
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From: Virgil on 30 Oct 2006 17:07 In article <1162217603.831099.28170(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162190663.531903.51440(a)e64g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Sebastian Holzmann schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > Sebastian Holzmann schrieb: > > > > > > > > > >> Virgil <virgil(a)comcast.net> wrote: > > > > >> > In article > > > > >> > <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, > > > > >> > mueckenh(a)rz.fh-augsburg.de wrote: > > > > >> >> What you propose, namely the infinity of ZF without the axiom INF > > > > >> >> would > > > > >> >> not be an advance. But meanwhile you may have recognized that > > > > >> >> your > > > > >> >> assertion (ZF even without INF is not finite) is false. > > > > >> > It is, however, quite true that ZF without INF need not be finite. > > > > >> It is, more than that, quite true that ZF without INF _is_ infinite > > > > > Are you really sure? > > > > > > > > I am. > > > > > > > So you also must be sure that ZF is yellow. > > > > And WM is equally sure that what natural numbers which he allows might > > exist are all purple with pink polka dots. > > > > > > >> (the axiom schema of separation alone provides infinitely many > > > > >> axioms). > > > > > Are you really sure? > > > > > > > > I am. > > > > > > But most of them are very dirty? > > > > > > > > >> The point is: ZF without INF does not prohibit the existence of > > > > >> infinite > > > > >> sets, nor does it force them to exist. > > > > > > > > > > It prohibits to speak of infinite sets and to recognize such sets. So > > > > > one cannot be sure, but you are? > > > > > > > > Given any model of ZF-INF, one cannot be sure if there are infinite > > > > sets. > > > > > > In particular the sets consisting of infinitely many rabbits would be > > > of great interest. > > > > And like most of WM's notions, such sets would be hare today and gone > > tomorrow. > > Get it right: It is nonsense to talk about infinite sets if there is no > axiom of infinity and, therefore, no possible definition of infinity. One can define what it means for a set to be infinite without having any example of one which is infinite. Dedekind's definition, for example, does not require an instanciation do be a validly constructed definition. So WM is talking nonsense when he says we cannot do what we are a actually doing. > "Given any model of ZF-INF, one cannot be sure if there are infinite > sets" is as silly an argument as talking about sets of rats or > rabbits, or sets being clean or dirty. It is WM's ideas about what is silly which are silly. Talking about whether sets can be infinite in ZF- INF is a good deal less silly that WM's delusions are.
From: Lester Zick on 30 Oct 2006 17:09 On 30 Oct 2006 12:12:00 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Get it right: It is nonsense to talk about infinite sets if there is no >> axiom of infinity and, therefore, no possible definition of infinity. > >No, YOU need to get it right. You have it completely wrong. We don't >need the axiom of infinity to define the predicate 'is infinite'. But you certainly need something you ain't got besides the adjective "infinite" to define the predicate "infinity". > We >need the axiom of infinity to prove that there is an object of which >that predicate holds. ~v~~
From: Virgil on 30 Oct 2006 17:09 In article <1162217788.219300.129420(a)e64g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sebastian Holzmann schrieb: > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > Sebastian Holzmann schrieb: > > >> mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > >> > Sebastian Holzmann schrieb: > > >> >> It is, more than that, quite true that ZF without INF _is_ infinite > > >> > Are you really sure? > > >> > > >> I am. > > >> > > > So you also must be sure that ZF is yellow. > > > > > >> >> (the axiom schema of separation alone provides infinitely many > > >> >> axioms). > > >> > Are you really sure? > > >> > > >> I am. > > > > > > But most of them are very dirty? > > > > Oh, they are all very similar to each other (but distinct non the less). > > "dirty" is a property which without the axiom of dirt is as well > defined as "infinite" without the axiom of infinity. "Dirty" is a property inherent in WM's "logic".
From: Lester Zick on 30 Oct 2006 17:11 On Mon, 30 Oct 2006 16:43:03 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Han de Bruijn wrote: >> David Marcus wrote: >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>mueckenh(a)rz.fh-augsburg.de wrote: >> >>>stephen(a)nomail.com schrieb: >> >>> >> >>>>Can you describe a continuous version of the problem where each >> >>>>"unit" of water has a well defined exit time? A key part of >> >>>>the original problem is that the time at which each ball is >> >>>>removed is defined and reached. This is crucial to the problem. It is >> >>>>not just a matter of rates. If you added balls 1-10, then 2-20, >> >>>>3-30, ... but you removed balls 2,4,6,8, ... then the vase is >> >>>>not empty at noon, even though the rates of insertions and removals >> >>>>are the same as in the original problem. So you cannot just >> >>>>say the rate is 10 in and 1 out and base an answer on that. >> >>> >> >>>The answer for any time t *before noon* is independent of the chosen >> >>>enumeration of the balls. Doesn't that fact make you think a bit >> >>>deeper? >> >> >> >>And add this to the fact that noon and beyond cannot exist >> >>in this problem. >> > >> > Are you still doing physics, water, and a finite number of molecules? >> > Let us know when you switch to mathematics. >> >> I'm DOING mathematics. Mathematics is NOT independent of Physics. > >I guess your definition of "mathematics" is different from mine. I think the only relevant question is whether your definition of "mathematics" is demonstrably true? If not I don't see that any definition of mathematics is more virtuous one way or the other. ~v~~
From: Virgil on 30 Oct 2006 17:12
In article <1162218523.435187.296410(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > There is no computable list > > > > > > of computable numbers. > > > > > > > > > > Why do you insist on this obvious fact? It does not support your > > > > > position. There is a list of all finite constructions or definitions > > > > > (encoded by numbers, G?del). This is the definition of countablity. > > > > > The diagonal number of this list shows the listed numbers are > > > > > uncountable. Contradiction. > > > > > > > > > > > > There is indeed a list of all possible finite constructions or > > > > definitions. Call this list A. However, this list must contain things > > > > that look > > > > like definitions but are not because the method given to produce > > > > a number does not halt. We cannot get a diagonal number from A, > > > > because some of the members of A do not give numbers. > > > > So there is no contradiction. > > > > > > A contains all finite words (construction fromulas, theorems). The > > > diagonal (cannot be longer than the lines and hence) is a finite word > > > too. That is enough to obtain a contradiction. > > > > > > A also contains a number of things that look like contruction formulas > > but are not (because they don't halt). > > Forget about Turing. It is not of interest here. Forget about mueckenh(a)rz.fh-augsburg.de nothing it says is of any use. |