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,
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:
-Loeb's theorem; (just this one makes the travel worth!)
-Muravitski & Kusnetsov, Boolos, Goldblatt theorems;
-Other theorems by Goldblatt
-Still Other theorems by Goldblatt.
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.