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

Date: Sat Apr 28 10:42:10 2001

Hello George,

I hope my last post was not to short. Don't be anxious when

I use the word "easy" about exercices. It really means "short"

and it means that one day you will find that easy. "Easyness"

is relative to familiarisation, which need some amount of time,

of course.

Don't hesitate to ask question, and any type of question.

(This remark applies also to any everythinger who follows

this thread, obviously).

BTW I intent to solve the last exercices. At least the proof

of the soundness of most modal theories presented so far.

Now we have embarked ourself in a long story, because the proof

I promise has the curious feature that, like the peano curves

which goes everywhere in a square, it goes almost

everywhere in the fields of logics. My proof is short but

need a large view on apparently distinct results.

(And this is hardly astonishing given the largeness of our goal).

To be sure, modal logics can be seen as a kind of "amuse gueule",

a french name for a sort of appetizer. It is also something

we could have bypassed, but it will help us a lot for the

interview of the sound machine and its "guardian angel".

Actually we will need also

-Predicate logic, and arithmetics

-weak logics (intuitionist logic, quantum logic)

-Algebraic semantics of weak logics

-Kripke semantics of weak logics

Now the heart of the matter, to talk like Smullyan, will be

self-reference. Technically that means diagonalisations,

diagonalisations and diagonalisations all the way through.

And of course we will need a minimal amount of theoretical

computer science, including Church's thesis, the role of

\Sigma_1 sentences, etc.

(With the computationalist hypothesis it would have been

suspect that we don't talk on the science of computability).

Then the interview itself will begin. We can follow the historical

progress of that interview:

-Goedel's theorem;

-Loeb's theorem; (just this one makes the travel worth!)

-Solovay's theorem;

-Muravitski & Kusnetsov, Boolos, Goldblatt theorems;

-Other theorems by Goldblatt

-Still Other theorems by Goldblatt.

-Visser's theorem;

It is the theorem by Solovay which will make clear the relation

between provability logic and some modal logics.

Boolos, Goldblatt, Visser has found result which will make part

of our the translation of the UDA argument almost transparent.

For this reason, it will be needed to recall the essential of

the Universal Dovetailer Argument UDA.

The methodology consists really in

a translation of the UDA in the language of a sound Universal Turing

machine, and our proof-result is given by the answer given by

the machine, and by its guardian angel (its "truth theory").

I will probably not strictly follow the present line. The next post

will be the solution of the Kripke soundness of the modal theories.

After that, just to change our mind, I will perhaps digress on

diagonalisations and computations.

I doubt that a plenitude contemplator like you will not appreciate the

power of the diagonal, sometimes aptly called the most transcendantal

operation in the whole mathematics.

I'm open to any suggestion you could have.

And let us accept the idea that we have all the time to do that:

no need to hurry up.

I will also think about some books. Boolos 1993 is of course very

well suited. Goldblatt 1993 also.

Bruno

Received on Sat Apr 28 2001 - 10:42:10 PDT

Date: Sat Apr 28 10:42:10 2001

Hello George,

I hope my last post was not to short. Don't be anxious when

I use the word "easy" about exercices. It really means "short"

and it means that one day you will find that easy. "Easyness"

is relative to familiarisation, which need some amount of time,

of course.

Don't hesitate to ask question, and any type of question.

(This remark applies also to any everythinger who follows

this thread, obviously).

BTW I intent to solve the last exercices. At least the proof

of the soundness of most modal theories presented so far.

Now we have embarked ourself in a long story, because the proof

I promise has the curious feature that, like the peano curves

which goes everywhere in a square, it goes almost

everywhere in the fields of logics. My proof is short but

need a large view on apparently distinct results.

(And this is hardly astonishing given the largeness of our goal).

To be sure, modal logics can be seen as a kind of "amuse gueule",

a french name for a sort of appetizer. It is also something

we could have bypassed, but it will help us a lot for the

interview of the sound machine and its "guardian angel".

Actually we will need also

-Predicate logic, and arithmetics

-weak logics (intuitionist logic, quantum logic)

-Algebraic semantics of weak logics

-Kripke semantics of weak logics

Now the heart of the matter, to talk like Smullyan, will be

self-reference. Technically that means diagonalisations,

diagonalisations and diagonalisations all the way through.

And of course we will need a minimal amount of theoretical

computer science, including Church's thesis, the role of

\Sigma_1 sentences, etc.

(With the computationalist hypothesis it would have been

suspect that we don't talk on the science of computability).

Then the interview itself will begin. We can follow the historical

progress of that interview:

-Goedel's theorem;

-Loeb's theorem; (just this one makes the travel worth!)

-Solovay's theorem;

-Muravitski & Kusnetsov, Boolos, Goldblatt theorems;

-Other theorems by Goldblatt

-Still Other theorems by Goldblatt.

-Visser's theorem;

It is the theorem by Solovay which will make clear the relation

between provability logic and some modal logics.

Boolos, Goldblatt, Visser has found result which will make part

of our the translation of the UDA argument almost transparent.

For this reason, it will be needed to recall the essential of

the Universal Dovetailer Argument UDA.

The methodology consists really in

a translation of the UDA in the language of a sound Universal Turing

machine, and our proof-result is given by the answer given by

the machine, and by its guardian angel (its "truth theory").

I will probably not strictly follow the present line. The next post

will be the solution of the Kripke soundness of the modal theories.

After that, just to change our mind, I will perhaps digress on

diagonalisations and computations.

I doubt that a plenitude contemplator like you will not appreciate the

power of the diagonal, sometimes aptly called the most transcendantal

operation in the whole mathematics.

I'm open to any suggestion you could have.

And let us accept the idea that we have all the time to do that:

no need to hurry up.

I will also think about some books. Boolos 1993 is of course very

well suited. Goldblatt 1993 also.

Bruno

Received on Sat Apr 28 2001 - 10:42:10 PDT

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