> > David C. Ullrich wrote: >> On Thu, 13 Jul 2006 13:52:31 -0400, Hatto von Aquitanien >> <abbot@AugiaDives.hre> wrote: >> >> >email@example.com wrote: >> > >> >> In sci.math Aatu Koskensilta <firstname.lastname@example.org> wrote: >> >>> Hatto von Aquitanien wrote: >> >>>> For example it was recently pointed out that permutations of N >> >>>> result in a demonstration that there exist uncountable bijections. >> >> >> >>> Really? Where can I find out more about this startling discovery? >> >> >> >> The above is Aquitanien's garbled interpretation of the fact >> >> that there exist countable sets that are not recursively >> >> enumerable. >> >> >> >> Stephen >> >If you cannot recursively enumerate it, you can't count it. >> >> That depends on exactly what you mean by "count it". And it >> has no relevance at all to the fact that there exist countable >> non-re sets (luckily "countable" does have a precise definition). > > I was taught that a set is "countable" if there exists a one-to-one > correspondence between the elements of the set and the natural > (counting) numbers.
Long before David's troll I had already retracted my statement. In the case of finite systems, there is no consequential distinction between the concepts of recursive enumerability. What had been stated is that there are 1-1 mappings to the natural numbers which are not recursively enumerable. Since I have grown very suspicious of the idea of manipulating the set of natural numbers using operations originally developed for finite sets, when I saw the claim that these mappings were countable and yet uncomputable, I over reacted in defense of rationality. I had failed to realize that the elements could be counted anonymously.
But as a general rule, I'm not going to accept definitions of terms such as countable which are not consistent with one cow, two cows, three cows... -- Nil conscire sibi