Hi Günther,
Nice work Günther. Now my comment is longer than I wish. I really
would insist on one change. See (**) below.
On 16 Feb 2009, at 22:54, Günther Greindl wrote:
>
> Hi guys,
>
> I finally got around to writing the AUDA references page:
>
> http://groups.google.com/group/everything-list/web/auda
>
> Comments welcome.
I would separate better the introduction to (general) mathematical
logic ...
Enderton (you mention it)
Mendelson (one of the best introduction to mathematical logic)
Perhaps the Podniek web page
The book by Boolos and Jeffrey (and Burgess for the last edition), and
the book by Epstein and Carnielli
Kleene's 1952 book on Metamathematics.
...from the general book on computability (but those books are really
needed already for the UDA, actually for the seventh step of UDA): so
I would put them there: I am thinking about
Cutland
Rogers
And then come the most fundamental books on the logic of self-
reference and/or provability logic per se (those are books on G and
G*). This is part of AUDA:
First the main initial original papers : Davis 1965 (contain Gödel
1931, Church, Post, Kleene, Rosser). Then the textbook on self-
reference (provability) logic:
Boolos 1979
Boolos 1993
Smorynski 1985
Smullyan's Forever undecided (a recreative introduction to the modal
logic G).
And then you can add some books on (general) modal logic (but they are
not needed because the book on provability logic reintroduces the
modal logic). You did already mentioned :
Chellas (excellent)
But the new edition of Hugues and Creswel is an easier one, and is
very good too imo.
The relation between modal logic and provability is a bit like tensor
calculus and general relativity. Modal logic is but a tool, provabilty
logic (sometimes called self-reference logics) is the object of study.
It is part of AUDA. "AUDA" really begins with Gödel's famous 1931
paper, and the very special modal logic G and G*, found by Solovay, is
a machinery encapsulating all the incompleteness phenomenon.
(**) If you want make just one little change in the page: in your
sentence "For modal logic these are further guides:" I would make
clear you are referring to the modal logic G and G*, that is the logic
of self-reference. Or just put "provability" or "self-reference"
instead of modal.
I would not put the Solovay paper in "guide on modal logic". It is
really the seminal paper on the self-reference logics.
The modal logic G and G* are really the logic of provability or self-
reference on which AUDA is based.
I am aware we touch "advanced matter", which presupposes a good
understanding of mathematical logic, or metamathematics, something
which is usually well known only by professional mathematical
logicians. Even a genius like Penrose got Gödel's wrong. By the way,
Hofstadter got Gödel's right in his book "Gödel, Escher, Back". He is
correct on computationalism too, but he missed the "matter problem",
and even the universal machine, the first person indetermincay and its
"reversal" consequences.
I have realized that some of my students have still a problem with
completeness and incompleteness. In part due to the bad choice in the
vocabulary (yet standard).
For example the theory PA (Peano Arithmetic) is complete in the sense
of Gödel 1930, and incomplete in the sense of Gödel 1931.
Completeness: (PA proves A) is equivalent with (A is true in all
models of PA). This makes "Dt" equivalent with "there is a reality":
the basic theological bet.
Incompleteness: there are true arithmetical statement (= true in the
standard model of PA) which are not provable by PA.
Don't hesitate to ask any question. Of course UDA is *the* argument.
AUDA is far more difficult and is needed to pursue the concrete
derivation of the physical laws (among all hypostases). UDA shows that
physics is a branch of computationalist self-reference logic. AUDA
begins the concrete derivation of physics from the existing self-
reference logic (thanks to Gödel, Löb, Solovay).
Note that for a time i have believed that the hypostases were all
collapsing. If this would have been the case, the comp-physics would
have been reduced to classical logic, and what we call physics would
have been a sort of comp-geography. The SWE would have been a local
truth.
Ask any question, we are in deep water. People like Tegmark and
Schmidhuber are on the right track concerning the ontology. The
intersection of Tegmark work and Schmidhuber's work gives the
"correct" minimal ontology: the mathematical elementary truth (on
numbers or mathematical digital machine). My (older) work derives this
from comp and the imperative of the mind body problem, which both
Schmidhuber and Tegmark seems not willing to take into account: they
presuppose some mind:machine identity which the UDA shows impossible
to maintain.
I cpntinue to think that for a non mathematician, a thorough
understanding of the UDA is needed before AUDA. UDA is really the
question, including the consequences that the solution has to be
given by the self-introspective universal machine; and AUDA is that
beginning of the universal machine's answer. For a logician AUDA is
far simpler than UDA, but only for them. My work, like the work by
Penrose illustrates that mathematical logicians are not well
understood by non logicians. Mathematical logicians lives in a ivory
tower.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Feb 18 2009 - 05:09:46 PST