[FOM] Godelian Semantic Domains ?

[A. Mani]

It is well known that Godel was a logician with primary interests in
mathematics this is directly connected to his version of mathematical realism.
By a formal semantic domain I mean a category with some reasonable
constraints on it ... so that we can do some model theory on it (as is done
in generalized or categorical abstract model theory).  It is not that the
latter is compulsory for the former (semantic domains can stand by
themselves). The aforesaid statement does suggest that corresponding to
Godel's perception, we may be able to define some semantic domains.

The other possible approach is via the institution-independent version (The
difference being in the concept of satisfaction being defined or axiomatized
and the level of classicalism involved).

I summarize the basic approach of generalized abstract model theory in
slightly modified terms ( *the types are brought in* ) so that it becomes
more suggestive of possible generalizations.  If A (= V^{A}_{0}) is a set,
then we can define the superstructures $V^{A}_{n}\,=\, V^{A}_{n-1} \cup
\wp{A}$ by induction and form $V^{A}_{\omega} = \bigcup _{n < \omega} V
^{A}_{n}$.
Given a multi-sorted type \tau in general, a pair of the form <A, p_{\tau}>
with p_{\tau} \in V^{A}_{\omega} is a generalized structure. This is formed
in the natural way.  (This way we can include most mathematical objects like
topological spaces ...). We can form the category of objects of type \tau,
with suitable morphisms. We can call these the basic categories.

A semantic domain C is a category consisting of generalized structures of
different types as objects and corresponding morphisms of type \tau formed by
categorical correspondences from the basic categories under the following
conditions ( The other approach is to see the semantic domain as a function
from types to generalized structures) : functor, closure, richness, finite
reduct, renaming, pair and diagram (see for e.g.  Mundici,  D.   A
generalization of abstract model theory, Fundamenta Mathematicae,  124 (1984)
pp. 1-25). Functor for example is intended to preserve identities and
composition across C and the basic categories, Closure is the requirement of
closedness of the objects of different types under the formation of reducts,
renaming, disjoint union, strict and pair expansions, etc.

Given a concept of *sentences of type \tau*, a Logic is then a triple of the
semantic domain, a satisfaction relation (to be defined) and the set of
sentences of different types. Of course more nicer conditions that guarantee
the existence of substructures are imposed on the semantic domains.

These semantic domains correspond to some domains of discourse in philosophy
(although to rigid mathematical versions). I think that they can be modified
suitably for settling philosophical disputes modulo some lesser disputes.
It is not that this is always easy. Take for example the philosophical
standpoint of finitism or ultra-finitism. If we modify the *richness*
condition in the concept of a semantic domain, then we are started.
*richness* in words means that from within the objects of the semantic domain
we can extract any object of type tau if that is involved in making the
semantic domain in the first place. Simply allow the extraction of ones that
are suitably finite. The finite reduct condition will also need to be
modified to *only for suitably finite subsets of \tau can we form reducts of
the structures*. The gains are in the generality of possible results.

I did check up some recent work on Godel, but could not find a formal version
of semantic domains that he may be said to have adhered to. The recent BSL
article (2005) by Martin Davis does suggest certain things but does not
disallow such a possibility in the sense that it is deducible that
Godel did not hold contradictory views at the same time in particular.

  But my feeling is that some work might have been already been done along
those lines.

Any pointers ?

Best

A. Mani
Member, Cal. Math. Soc


p.s. thanks are due to the editors for pointing out a difficulty with the
terminology.