Cantor Confusion
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From: David Marcus on 12 Oct 2006 16:43 Han de Bruijn wrote: > Dik T. Winter wrote: > > > 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. You state it is true for each finite > > sequence, so it is also true for the infinite sequence. That conclusion > > is simply wrong. > > That conclusion is simply right. And yours is wrong. Completed infinity > does not exist. So _each_ finite sequence "means" the infinite sequence. Are you saying that "completed infinity" does not exist in standard mathematics? If so, please define "completed infinity". If not, then please define "exist". -- David Marcus
From: David Marcus on 12 Oct 2006 16:47 Han de Bruijn wrote: > Dik T. Winter wrote: > > 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. > > Set Theory is simply not very useful. The main problem being that finite > sets in your axiom system are STATIC. They can not grow. Which is quite > contrary to common sense. (I wouldn't imagine the situation that a table > in a database would have to be redefined, every time when a new row has > to be inserted, updated or deleted ...) Is your claim only that set theory is not useful or is contrary to common sense? Or, are you claiming something more, e.g., that set theory is mathematically inconsistent? -- David Marcus
From: Alan Morgan on 12 Oct 2006 17:15 In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Dik T. Winter wrote: > >> 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. You state it is true for each finite >> sequence, so it is also true for the infinite sequence. That conclusion >> is simply wrong. > >That conclusion is simply right. And yours is wrong. Completed infinity >does not exist. sqrt(-1) doesn't exist either. Frankly, I have a much harder time believing in "imaginary" numbers than I do believing in infinite sets. Alan -- Defendit numerus
From: Virgil on 12 Oct 2006 17:45 In article <1160675643.344464.88130(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > With the diagonal proof you cannot show anything for infinite sets. Maybe "Mueckenh" can't but may others can.
From: Virgil on 12 Oct 2006 17:56
In article <1160676113.344404.246370(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Cantor uses Lim{n} n = omega witout much ado. > > > > That is not a limit of "sets". > > It is the limit of the natural numbers. The limit is omega, an ordinal > number. Meanwhile we know that every number is a set. Hence, it is a > limit of sets. N ( or omega) is only a limit in the sense of being a union of its members, and is the first non-empty ordinal to be equal to the union of its members. No other form of 'limit of a sequence' of sets is defined in ZF. > > > > > In his first paper he uses even Wallis' symbol oo. What should there > > > require a definition, if all natural did exist? > > > This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} > > > = N. > > > > Yes, and that fits my definition. On the other hand, how would you > > define lim{n --> oo} {n, n+1, ...}? > > I would not attempt to define that. It is obviously the empty set, i.e., the intersection of all the sets of form {n, n+1, ...}. lim{n --> oo} {n, n+1, ...} = lim{n --> oo} N\{0,..,n-1} = N\(lim{n --> oo} {0,..,n-1}) = N\ Union({n:n in N}) = N\N = {} |