It is good. Not as far as us, but this avoids the common mistake when
using Incompleteness outside logic. Of course the author seem ignore
(there is no references) that by going (quite) a bit farer, we get an
explanation of consciousness (true, knowable, undoubtable,
undefinable, and unjustifiable) *and* of the origin and shape of the
physical laws, making Digital Mechanism testable.
What still many miss, but that Post intuited, and Webb and Kleene
understood largely, is the unavoidable conflict between all the ways
of approaching truth enforced by incompleteness to the machine
reasoning about itself or selves (1p, 3p, ...). Indeed, incompleteness
forces the nuances between the 8 (!) nuances of "beweisbar":
p ((sigma) truth)
p (rationally justifiable, provable ("beweisbar" itself, when
translated in the arithmetical language)
p & p (first person, knowable)
p & ~f ("bet-table", predictable, observable, distinguishable)
p & ~f & p (sensible yet partially measurable, qualia)
The illumination, which is also the blasphemy if presented as being
true, is that
G1* proves p <-> p <-> (p & p) <-> (p & ~f) <-> (p & ~f &
but the Löbian, terrestrial, effective condition is that G does not
prove any of those equivalence, and the Löbian entity knows it
(machine and non-machine).
G1 proves p -> p, indeed, it is taken as axiom, as it characterizes
in particular the machine aware of their own Turing universality. "p -
> p" expresses "Turing universality" in arithmetic in some weak
technical sense: they can prove for any sigma_1 sentence that if they
are true then they can prove it: that is for each arithmetical
sentence p, they can prove p -> Beweisbar('p'), modally "p -> p".
But the machine itself, as defined correctly (hopefully) by its
susbtitution level and personal relative description, is literally
enforced to see the truth (p) "masked" by the body (p). The Löbian
entity can't prove p -> p, given that she can't already prove, even
taken as an axiom, f -> f, that is its own consistency (~f). And
this makes all the other nuances obeying to different logics.
It match well, and that is plausibly not a coincidence, the discourse
of those open to the mystical experience and rationalism like the
neopythagoreans and neoplatonists.
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