An uncountable countable set
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From: Virgil on 13 Aug 2006 13:27 In article <1155485913.329014.192160(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > So Cantor's diagonal is never completed. We are never sure that it is > not in the list. Everyone but "Mueckenh" is.
From: Virgil on 13 Aug 2006 13:30 In article <1155486119.103774.163800(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <J3uzrx.Apr(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > > In article <1155315217.378759.114420(a)b28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > Also you state that indexing is the same as covering. This is the > > > same as telling: > > > for all p there is an n such that ... > > > is equivalent to: > > > there is an n such that for all p ... > > > which is false. > > > > > > Back again to quantifier dyslexia. > > > > For a simple (finite) example where quantifier dyslexia tells the wrong > > things can be found at: > > <http://mathworld.wolfram.com/EfronsDice.html> > > let p and n be (different) dice numbers (from 1 to 4) the first statement > > would be: > > for all p there is an n such that ... > > the second would be: > > there is an n such that for all p ... > > where ... stands for the statement "the probability of throwing higher > > with die n than with die p is larger than the reverse". The first is true, > > the second is false. You pick one first, next I pick one, and I have a 2:1 > > chance of winning when we throw, regardles of the one you pick. > > > > When you look around a bit more there are many more interesting examples. > > There is even one where the situation is clear with a set of three dice > > of different colours, but where it reverses if you duplicate the set > > and start throwing with two dice of the same colour. > > That is all very interesting and very ------------------- misleading. > > For a linear problem like the unary numbers of the list, there is no > quantifier gambling possible. If position n is indexed by a unary > number, then also all positions m < n are covered by this number n. > There is no outcome in the set of numbers of the form 0.111...1. > > 1) If a digit is indexed then the sequence up to that digit is covered. > 2) If there are no digits which cannot be indexed, then there are no > digits, which cannot be covered. > 3) That is equivalent to the statement, that all digits can be indexed > and all digits are covered. Which is quite valid in ZF > > You assert that every digit of 0.111... can be covered, but that > 0.111... cannot be covered. In fact, by the Hilbert Hotel method, one can cover any finite number of extras after each digit and the whole are covered. That assertion is void of meaning. It makes > no sense. And examples of dice or of all women who have men and of a > woman who has all men do not apply to this case. > > Regards, WM
From: Virgil on 13 Aug 2006 13:32 In article <1155486317.573725.325460(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > That is a special formalization which was given long, long after the > limit had been in use. One of the first limit-users was Archimedes when > exhausting the area of the parable. I though it was Aesop who did the parables.
From: Virgil on 13 Aug 2006 13:36 In article <1155486471.373626.282060(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > Do you know how the infinite list of Cantor can be enumerated? > > > > No, of course not. It is not the list of Cantor, but the list provided > > to Cantor by somebody else. I hope that the person that provided the > > list did know how the list was enumerated. > > Nonsense. Cantor invented the list and constructed the first one. Lists, in the form of sequences, were around long before Cantor, so "Mueckenh" is wrong again. > > > > > Do you know how the infinite set of algebraic numbers can be > > > enumerated? > > > > Yes. > > And why did you believe that the set of edges of my tree was > uncountable? No one does, But the set of paths in an infinite binary tree is. > > Regards, WM
From: Virgil on 13 Aug 2006 13:37
In article <1155486556.890586.315500(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Therefore it cannot be indexed. Every digit position which can be > indexed is in the true list. Hilbert could do it, but "Mueckenh" is a good deal less capable. |