On Jun 17, 1:49 am, Tim Little <t...@little-possums.net> wrote: > On 2010-06-17, Peter Webb <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > > "Virgil" <Vir...@home.esc> wrote in message > >> An infinite set is defined to bee countable if and only if there is a > >> surjection from the set of natural numbers to that set. When such a > >> function is a bijection, it is commonly called a list. > > > Only if the bijection can be explicitly created. > > You apparently have some bizarre private definition of "list". > > Explicitness has nothing to do with it. > > Though even if it did, you are incorrect. Given any numbering of > Turing machines, a list of computable reals ordered by the > least-numbered Turing machines that compute them is quite explicit. > > The practical difficulties of establishing which Turing machines halt, > which are equivalent and so on are just that: practical difficulties > which have nothing to do with mathematical theory. > > - Tim
Uncountability of the rationals? The rationals are well known to be countable, and things aren't both countable and uncountable, so to have a reason to think that arguments about the real numbers that are used to establish that they are uncountable apply also to the rationals, the integer fractions, has for an example in Cantor's first argument, about the nested intervals, that the rationals are dense in the reals, so even though they aren't gapless or complete, they are no- where non-dense, they are everywhere dense on the real number line. So, even though the limit of the sequences that are alternatively sampled to bound the minimum might not be in the rationals, to be an irrational number in the reals, still at every step there are more rationals then between the limiting value defined by the function from the naturals to the real numbers (or rational number). Between them are some non-rational numbers, the irrational numbers, not integer fractions although everywhere sums of them with infinite sums like Cauchy sequences besides the finite, the rationals are not continuous anywhere, although the normal definition of continuity as is presented includes the rationals. Many of the arguments that result from normal definitions of continuity apply to the rational numbers which in analysis are generally ignored in deference to the real numbers. That's still the general result of approximation, here the function that defines the evolution of the minimum bounds defined upper and lower by the two sequences of the evens and odds of F(n) as defined preserves for example Markov sequence, error term, ergodicity, of course any results. Also preserves there, the function, looking at all the functions from the natural integers to the real numbers, besides of course any results, neat translations from [0,1] to for example [-1/2,1/2], or, [-1,1], i.e., average zero.