On 05 Jun 2016, at 01:38, Russell Standish wrote:
On Sat, Jun 04, 2016 at 07:53:47PM +0200, Bruno Marchal wrote:
On 04 Jun 2016, at 03:17, Russell Standish wrote:
Only a measure on the existing computations.
Of course, all this list is based on the idea that the overall
theory should be simple (like RA), but even without notion of
observers, such simple theory admit a rich third person theory of
complexity.
I still don't see it.
You can prove the existence of the observer in the theory RA. It is
the proof of the existence of the universal machine and of their
computations. RA is already sigma_1 complete. If ZF proves the
Riemann hypothesis, then RA will prove that ZF proves Riemann
hypothesis (despite RA itself cannot even prove that 0 + x = x).
We don't need to assume more than RA or the SK axioms. 3p Observers
are defined in such theories. They are richer and more complex than
RA.
This is close to the crux of out disagreement. Yes of course by
assumption observers can be defined in RA by finding the right
machine. However, what the observers observe is not defined in RA, but
rather emerges out of RA.
OK.
I see that RA itself is a fairly simple thing, and the UD even simpler
for that matter, but emerging out of it are 1p experiences that are
complex, and complexify over time via an evolutionary process.
We should distinguish the evolution proper to this or that
computational3p-sharable histories, and the "evolution" by the FPI
selection. If not we will have problem to distinguish first person
singular and first person plural. We would lose the ability to justify
the existence of such first person plural reality from the first
person singular reality and fall into solipsism. Fortunately, the self-
reference logics forces the distinction all by itself.
The complexity is not innate to the platonic realm as you state. It
emerges out of it by considering 1p experience.
We have the two type of complexity, and they are of very different
nature. Once is brought by the usual 3p arithmetic complexity, or
computer science, the other comes from infinite limit on infinitely
many computations, structured by the logic of self-reference.
My fairly strong claim is that this is the only way complexity can
arise.
The complexity of the partition function in number theory, or of the
complexity of the distribution of the primes is there already in all
models of RA or PA, and constitute a 3p-complexity studied by
mathematicians (having usually no interest in modalities like proof,
knowledge, observation, 1p-sharable or not). G and G* are still 3p,
and at the quantifier level (first order modal logic) they are very
complex. qG is PI_2 complete, and qG* is PI_1 complete in the oracle
of the arithmetical truth itself.
Bruno
Cheers
--
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Dr Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow [email protected]
Economics, Kingston University http://www.hpcoders.com.au
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