On 17 May 2015, at 20:24, meekerdb wrote:
On 5/17/2015 1:44 AM, Bruno Marchal wrote:
Roughly speaking, consciousness originates from the fact that p ->
[]p (the sigma_1 truth get represented in the body/brain of the
machine), and the fact that []p -> p is true, but non justifiable
by the machine.
What does []p mean?
An abbreviation of beweisbar('p'). It is Gödel arithmetical
provability predicate, and we have an equivalent for any machine
talking correctly about itself, seen in the 3p picture.
p doesn't entail that p is provable or necessary.
It does, when p is restricted to sigma_1 sentences. Sigma_1 means
equivalent to a sentence with the shape ExA(x, ...), with A(x, ...)
decidable.
You can understand that if A(x, ...) is decidable, you will find the x
verifying it, by testing
A(0, ...)
A(1, ...)
A(2, ...)
etc. Until you find it.
When ExA(x, ...) is true (p) then it is provable (by any sigma_1
complete machine, like RA, and this is equivalent with Turing
universality). So p -> []p.
You might have just skipped that we limit ourself to the sigma_truth.
All such sigma_1 truth is provable by the machine.
Löbian machine are not only sigma_1 complete (and thus p -> []p is
true for them, with p Sigma_1), but they can prove "p-> []p", for any
p sigma_1. In particular []p is sigma_1, so we get the self-awareness
principle: []p -> [][]p.
Ask any question if this is not clear enough.
Bruno
Brent
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