Cantor Confusion
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From: Han.deBruijn on 14 Oct 2006 16:32 David Marcus schreef: > Han de Bruijn wrote: > > > > I'm not interested in the question whether set theory is mathematically > > inconsistent. What bothers me is whether it is _physically_ inconsistent > > and I think - worse: I know - that it is. > > What does "physically inconsistent" mean? Wouldn't your comments be > better posted to sci.physics? Most people in sci.math are (or at least > think they are) discussing mathematics. ONE world or NO world. Han de Bruijn
From: mueckenh on 14 Oct 2006 16:32 Dik T. Winter schrieb: > In article <1160675932.010700.124010(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > It is not > > > > contradictory to say that in a finite set of numbers there need not be > > > > a largest. > > > > > > It contradicts the definition of "finite set". But I know that you are > > > not interested in definitions. > > > > We know that a set of numbers consisting altogether of 100 bits cannot > > contain more than 100 numbers. Therefore the set is finite. The largest > > number of such a set cannot be determined, as far as I know. > > That set is indeterminate. Just use Ascii notation. The string > "Graham's number" fits in 100 bits. > > > Could you determine it? Or would you prefer to define that such ideas > > do not belong to mathematics? Then I would not be interested in that > > definition. > > It is an indeterminate set. The set of bits is determined: exactly 100. What you can build from 100 bits belongs to the power set of this set. It is probably large but certainly not infinite. Regards, WM
From: mueckenh on 14 Oct 2006 16:34 Dik T. Winter schrieb: > In article <1160675848.377420.163220(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > If every digit position is well defined, then 0.111... is covered "up > > > > to every position" by the list numbers, which are simply the natural > > > > indizes. I claim that covering "up to every" implies covering "every". > > > > > > Yes, you claim. Without proof. > > > > 1) "Covering up to n" > > means > > 2)"covering n" > > and "covering the predecessors of n". > > Therefore we need not prove (2) if (1) is true. > > Yes. But you claim: (3) "covering 0.111...", not covering n. Yes. But you claim 0.111... consists merey of finite n. That is the error. "...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Weyl) > > > You state it is true for each finite > > > sequence, so it is also true for the infinite sequence. > > > > It is true *for every finite position*. I do not at all care how many > > such positions there are. The obvious covering of (2) by (1) does not > > depend on frequency. > > But were is the "covering up to 0.111..."? If there are actually infinitely many positions, then 0.111... is not completely covered, hence not defined , hence not existing, hence the "if there are actually infinitely many positions" contradicts itself. Therefore we have only potential infinity. There is every number of digits you can conceive of, but it is never actually infinite. Regards, WM
From: Virgil on 14 Oct 2006 16:36 In article <1160814355.022151.205320(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Your assumption that some ball that has been removed has not been > > removed is even more absurd. > > Yes, that is true. Therefore the existence of all natural numbers can > be excluded. To assume it is absurd. In proofs by reductio ad absurdum, one must be quite careful what one claims is the absurdity, as it is easy to mistake which claim is the absurd one. And "Mueckenh" regularly makes such mistakes.
From: mueckenh on 14 Oct 2006 16:38
William Hughes schrieb: > > You think we cannot name the M, so it cannot exist? You know we cannot > > construct a well-order of the reals. But nevertheless it does exist, > > according to Zermelo's proof. And here, there *is* such an M, for all > > finite positions, according to a very simple proof, independent of > > whether or not we can name it. > > > No. We do not have one M. We have a lot of > different M(N)'s, one for each digit position. And each one covers all it predecessors. > We can > show that if there is a last N, call it K, then all > the M(N) can be replaced by a single M(K). Exactly. > But there > is a last N iff the number of digit positions in 0.111.... > is finite. Exactly. Either the number of digit positions is finite or there are some positions undefined or an infinite set does not actually exist, meaning that the number of digits is finite though unbounded. These are the alternatives. The big mistake of ZFC is to exclude the last alternative. Remember this crank: "...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Weyl) Regards, WM |