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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Thu, 1 Sep 2005 15:54:40 +0200

On 31 Aug 2005, at 17:11, kurtleegod.domain.name.hidden wrote:

*> This I don't quite follow. Sorry! How are "conditions of
*

*> observability" defined by CT?
*

This is obviously technical, but in a nutshell (see more in the

papers):

By the UD Argument (UDA, Universal Dovetailer Argument), we know,

assuming comp, that all atomic or primitive observer moment

corresponds to the states accessible by the Universal Dovetailer (CT

is used here). This can be shown (with CT) equivalent to the set of

true *Sigma_1 arithmetical sentences* (i.e those provably equivalent,

by the lobian machines, to sentences having the shape EnP(n) with P

decidable. For a lobian machine, the provability with such atomic

sentences is given(*) by the theory G + (p -> Bp). Now, a

propositional event will correspond to a proposition A true in all

accessible observer-moments (accessible through consistent

extensions, not through the UD!). And this in the case at least one

such accessible observer-moments exists (the non cul-de-sac

assumption). Modally (or arithmetically the B and D are the

arithmetical provability and consistency predicates), this gives BA &

DA. This gives the "conditions of observability" (as illustrated by

UDA), and this gives rise to one of the 3 arithmetical quantum logic.

The move from Bp to Bp & Dp is the second Theaetetical move. Dp is

~B~p. Read D Diamond, and B Box; or B=Provable and D=Consistent, in

this setting (the interview of the universal lobian machine). Part of

this has been motivated informally in the discussion between Lee and

Stathis (around the "death thread"). Apology for this more "advanced

post" which needs more technical knowledge in logic and computer

science.

Bruno

(*) EnP(n) = it exists a natural number n such that P(n) is true. If

p = EnP(n), explain why p -> Bp is true for lobian, or any

sufficiently rich theorem prover machine. This should be intuitively

easy (try!). Much more difficult: show that not only p -> Bp will be

true, but it will also be *provable* by the lobian machine. The first

exercise is very easy, the second one is very difficult (and I

suggest the reading of Hilbert Bernays Grundlagen, or Boolos 1993, or

Smorinsky 1985 for detailled explanations).

PS: I must go now, I have students passing exams. I intent to

comment Russell's post hopefully tomorrow or during the week-end.

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

Received on Thu Sep 01 2005 - 10:03:03 PDT

Date: Thu, 1 Sep 2005 15:54:40 +0200

On 31 Aug 2005, at 17:11, kurtleegod.domain.name.hidden wrote:

This is obviously technical, but in a nutshell (see more in the

papers):

By the UD Argument (UDA, Universal Dovetailer Argument), we know,

assuming comp, that all atomic or primitive observer moment

corresponds to the states accessible by the Universal Dovetailer (CT

is used here). This can be shown (with CT) equivalent to the set of

true *Sigma_1 arithmetical sentences* (i.e those provably equivalent,

by the lobian machines, to sentences having the shape EnP(n) with P

decidable. For a lobian machine, the provability with such atomic

sentences is given(*) by the theory G + (p -> Bp). Now, a

propositional event will correspond to a proposition A true in all

accessible observer-moments (accessible through consistent

extensions, not through the UD!). And this in the case at least one

such accessible observer-moments exists (the non cul-de-sac

assumption). Modally (or arithmetically the B and D are the

arithmetical provability and consistency predicates), this gives BA &

DA. This gives the "conditions of observability" (as illustrated by

UDA), and this gives rise to one of the 3 arithmetical quantum logic.

The move from Bp to Bp & Dp is the second Theaetetical move. Dp is

~B~p. Read D Diamond, and B Box; or B=Provable and D=Consistent, in

this setting (the interview of the universal lobian machine). Part of

this has been motivated informally in the discussion between Lee and

Stathis (around the "death thread"). Apology for this more "advanced

post" which needs more technical knowledge in logic and computer

science.

Bruno

(*) EnP(n) = it exists a natural number n such that P(n) is true. If

p = EnP(n), explain why p -> Bp is true for lobian, or any

sufficiently rich theorem prover machine. This should be intuitively

easy (try!). Much more difficult: show that not only p -> Bp will be

true, but it will also be *provable* by the lobian machine. The first

exercise is very easy, the second one is very difficult (and I

suggest the reading of Hilbert Bernays Grundlagen, or Boolos 1993, or

Smorinsky 1985 for detailled explanations).

PS: I must go now, I have students passing exams. I intent to

comment Russell's post hopefully tomorrow or during the week-end.

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

Received on Thu Sep 01 2005 - 10:03:03 PDT

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