On 19/06/2010 11:43 AM, |-|ercules wrote: > "Virgil" <Virgil@home.esc> wrote >> In article >> <email@example.com>, >> WM <firstname.lastname@example.org> wrote: >> >>> Pi is constructable and computable and definable, because there is a >>> finite rule (in fact there are many) to find each digit desired. But >>> as there are only countably many finite rules, there cannot be more >>> defined numbers. >> >> If there are countably many rules then there are uncountably many >> lists of rules capable of generating a number. >> >> >> >> >> >>> Therefore matheologicians have created undefinable >>> "numbers". >> >> WM mistakes the issue. >> In pure mathematics, like in games, one has a set of rules to follow. >> Differing sets of rules generate differing systems only some of which >> are of much mathematical interest. >> >> The systems of rules we chose to use need not be subject to the >> constraints that the system of rules that WM choses to play by are >> subject to. >> >> For example, in FOL+ZFC, a commonly used system of rules which WM doe >> not care for, all sorts of things are legitimate that none of WM's >> systems of rules will allow. >> >> WM tries to force everyone to play only by his rules, but most of us >> find his system of rules dead boring and of little or no mathematical >> interest. >> Fortunately, outside of those classrooms in which his poor students >> are compelled to play by his rules, he has no power to impose those >> rules on anyone. > > Unfortunately your last paragraph is speculation and your superinfinity > rules > are full of contradictions. > > You show this example as Cantor's method > > 123 > 456 > 789 > > Diag = 159 > AntiDiag = 260 > > But this method miserably fails on the computable set of reals, THERE IS > NO new sequence like 260 in this example.
Computable set of reals? Set of computable reals?
Cantor proves that the set of reals is not countable. It's not saying anything about computables.
> > Why don't you rework Cantor's proof to define ALL anti-diagonals instead > of 1 particular anti-diagonal?
Because there's no need. We hyphothesise that the reals are countable. Since they are countable we can list them, and the list will contain all the reals. Then we show that there's a number that isn't in the list, which contradicts the hypothesis, which must therefore be wrong. Hence the reals aren't countable. It seems straightforward enough.
To avoid this proof, you have to dispute reductio-ad-absurdum arguments in general, argue that reals cannot be expressed as infinite strings of digits, or have axioms that exclude the construted anti-diagonal from the set of reals.