"Sylvia Else" <email@example.com> wrote... > On 19/06/2010 11:43 AM, |-|ercules wrote: >> "Virgil" <Virgil@home.esc> wrote >>> In article >>> <firstname.lastname@example.org>, >>> WM <email@example.com> 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?
I mixed up "miserably fails" with "fails miserably" but don't see what your distinction is here.
> > 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.
Do it anyway and you'll see what a bogus argument you have!!
> 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. > > Sylvia.
To support your argument you should at least show that you've formed a new sequence of digits.
If you actually read my derivation of herc_cant_3 instead of blindly dismissing it, you'll see it holds, just like all digits of PI appear in order below this line, if interpreted correctly.