ROADMAP (hypostases)

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Sun, 20 Aug 2006 15:59:38 +0200

Let me think aloud,

Plotinus's terms:

Primary Hypostases:
  1) the ONE
  2) the Divine Intellect
  3) the all-soul
Secondary hypostases:
  4) Intelligible Matter
  5) Sensible Matter

With the UDA, you can already try

Primary Hypostases:
  1) truth
  2) third person communicable truth
  3) first person truth
Secondary hypostases:
  4) probability on computationnal consistent states/histories
  5) probability on computational consistent true states/histories

With the lobian interview the self-referential correct intellect is
given by the modal logic G, and the self-referential truth (including
the non provable one) is given by G*. This gives the following
interpretation of a weaker version of UDA in arithmetic (comp is not
yet needed); the hypostases are with B for Godel's purely arithmetical
provability predicate (Beweisbar):

Primary Hypostases:
  1) arithmetical truth (p)
  2) provability (Bp)
  3) provability-and-truth (Bp & p)
Secondary hypostases:
  4) provability-and-consistency (Bp & ~B~p)
  5) provability-and-consistency-and-truth (Bp & ~B~p & p)

But, thanks to incompleteness, and the fact that machine as rich as PA,
can reflect that incompleteness, some hypostases' discourses are
divided in two parts: the true, and the communicable (third person
provable) one. We get 8 hypostases:


Primary Hypostases:
  1) arithmetical truth (p)
  2) provability (G) -------- 2') the same, but described by G*
  3) provability-and-truth (S4Grz, curiously enough it does not divide)
Secondary hypostases:
  4) provability-and-consistency (Z)-------- 4') same, but described by
G* (= Z*)
  5) provability-and-consistency-and-truth (X)-------- 5') same, but
described by G* (X*)

Until now, we have not yet introduced comp in the interview.

With B = Beweisbar; comp can be translated by p -> Bp. This formula
characterized the Sigma1 formula (Visser Theorem), that is the RE sets,
the Wi, the accessible states by a Universal Machine (with CT).

Let V = G + (p -> Bp)

We get

Primary Hypostases:
  1) Sigma1 arithmetical truth (p)
  2) provability (V) -------- 2') the same, but described by G* (V*)
  3) provability-and-truth (S4Grz1, curiously enough it does not divide)
Secondary hypostases:
  4) provability-and-consistency (Z1)-------- 4') same, but described by
  G* (= Z1*)
  5) provability-and-consistency-and-truth (X1)-------- 5') same, but
described by G* (X1*)

The logical of the physical proposition should emerge at least in Z1*.
But actually the whole of S4Grz1, Z1*, and X1* define, at least
formally, a notion of arithmetical quantization.

Bruno


http://iridia.ulb.ac.be/~marchal/


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Received on Sun Aug 20 2006 - 10:01:47 PDT

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