[ExI] Trilemma of Consciousness

[Adrian Tymes]

On Thu, May 25, 2017 at 7:33 AM, Stuart LaForge <avant at sollegro.com> wrote:
> Axiom 1: Let F(n) be the nth computable function with n being an
> admissible numbering of all possible computable functions.
> Axiom 2: Let K be the subset of F(n) such that all K share a semantic
> property k.
> Definition 1: Let k be called trivial if all F(n) have property k.
> Definition 2. Let k be called null if no F(n) has property k.
> Axiom 3: Let Dt be the decision problem as to whether a given F(n) belongs
> in K.
>
> Theorem: By axioms 1-3, definitions 1-2, and Rice's Theorem, Dt is
> decidable if, and only if, k is trivial or null.

Counterproof:

Let k be, "Given the numbering in use, is n > 1?"  (This is only null
if n is at most 1, though in that case all ks will be either trivial
or null anyway.)  This appears to be computable.

Let n be some integer greater than 1.  (I am assuming the numberings
only considers integer numberings starting with 1, but this proof
works with only slight modification to the examples if not.)

k is neither trivial (since F(1) does not have this property) nor null
(since F(2) has this property, as are others if n > 2).

However, Dt is decidable: F(1) is not in K.  The rest, which consists
of at least F(2), are in K.