[Haskell-cafe] type metaphysics
Ryan Ingram
ryani.spam at gmail.com
Mon Feb 2 17:39:25 EST 2009
2009/2/2 Luke Palmer <lrpalmer at gmail.com>:
> However! If we have a function f : Nat -> Nat -> Bool, we can construct the
> diagonalization g : Nat -> Bool as: g n = not (f n n), with g not in the
> range of f. That makes Nat -> Bool "computably uncountable".
This is making my head explode. How is g not in the range of f?
In particular, f is a program, which I can easily implement; given:
compiler :: String -> Maybe (Nat -> Bool)
mkAllStrings :: () -> [String] -- enumerates all possible strings
I can write f as
f n = validPrograms () !! n
where
validPrograms = map fromJust . filter isJust . map compiler . mkAllStrings
Now, in particular, mkAllStrings will generate the following string at
some index, call it "stringIndexOfG":
<source code for compiler>
<source code for mkAllStrings>
<source code for f>
g n = not (f n n)
This is a valid program, so the compiler will compile it successfully,
and therefore there is some index "validProgramIndexOfG" less than or
equal to "stringIndexOfG" which generates this program.
But your argument seems to hold as well, so where is the contradiction?
-- ryan
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