1.3. Form of theories: notions, objects and meta-objects
The variability of the model
Each consistent theory assumes its model (interpretation) as fixed,
but this is usually a mere «choice» of one model in a wide (infinite)
range of other existing, equally legitimate models of the same
theory; the model becomes variable when viewed by model theory. But,
this «choice» and this «existence» of a model can be quite abstract.
In details, the proof
of the Completeness theorem, in the way it can work in all
cases, will effectively «specify» a model in the range of
possibilities, but this construction is not really explicit
(involving an infinity of steps, where each step depends on an
infinite knowledge). In these conditions, the assumption of fixation
of a model may be called nonsense, but nevertheless constitutes the
standard interpretation of mathematical theories.
Notions and objects
Each theory has its own list of notions, usually designated
by common names, that are the kinds of variables used by the theory
; each model (interpretation of the theory) interprets each notion as a
set that is the common
range of all variables of this kind. For example, Euclidean geometry
has the notions of «point», «straight line», «circle» and more. The
objects of a theory in a model, are all the possible values
of its variables (the elements of its notions) in this model.
When we discuss several theories T and systems M
that may be models of those T, we are in the framework of
model theory, with its notions of «theory» and «system» that are
the respective kinds of the variables T and M. But when
we focus the study on one theory (such as a set theory) with a
supposedly fixed model, the variables T and M become
fixed and disappear (they are not variables anymore, the choice of the
theory and the model becomes implicit). So, the notions of theory and
model disappear from the notions list too.
This fixation reduces the framework, from model theory, to that of
one-model theory. A model of one-model theory, is a system
[T,M] that combines a theory T with a model
M of T.
On the diversity of logical frameworks
Before giving a theory T, we must specify its logical
framework (its format, or grammar), that describes the admissible
forms of contents for T, what such contents mean about M,
and how their consequences can be deduced. This framework is given
by the choice of a precise version of one-model theory, that describes
T and interprets its claims.
We shall first describe two of the main logical frameworks in parallel.
Theories in the most common framework of first-order
logic, will be called here generic theories. Set
theory will be expressed in is own special framework. More
frameworks will be introduced in Part 3.
The most common logical frameworks except the special one
of set theory, will manage notions as types (usually in
finite number for each theory) classifying both variables and
objects: each object will belong to only one type, the one of
variables that can name it. For example, an object of Euclidean
geometry may be either a point or a straight line, but the same
object cannot be both a point and a straight line.
Examples of notions from various theories
||Kinds of objects (notions)
||Pure elements classified by types
||Elements, sets, functions, operations,
||Theories, systems and their components
(listed next line)
| One-model theory
||Objects, symbols, types, structures,
expressions (terms, formulas)...
||Points, staight lines, circles...
The notions of a one-model theory T1, normally
interpreted in [T,M], classify the components of T
(«type», «symbol», «formula»...), and those of M («object»,
and tools to interpret T there). But the same notions (even if from
a different version of one-model theory) can be interpreted in
[T1, [T,M]], by putting the prefix meta- on them.
By its notion of «object», one-model theory distinguishes the
objects of T in M among its own objects in [T,M],
that are the meta-objects. The above rule of use of the meta prefix
would let every object be a meta-object; but we will make a
vocabulary exception by only calling meta-object those which
are not objects: symbols, types (and other notions), structures,
Set theory only knows the ranges of some of its own variables, seen
as objects (sets). But, seen by one-model theory, every variable of
a theory has a range among notions, which are meta-objects only.
Components of theories
Once chosen a logical framework, the content of a theory (or its foundation,
i.e. its initial content, describing the form of its intended
models), consists in a choice of 3 successive lists of components,
where those in each list are used to build those of the next list:
- A list of abstract types to serve as the names of types;
- A language (vocabulary): list of structure
symbols, names of structures relating objects to form systems (see 1.4).
- A list of axioms (among ground formulas of the
theory, see 1.5.).
Any generic theory (and its model, if considered) can be inserted
(translated) into set theory by converting its components into
components of set theory. Let us present both the generic method
which works for any generic theory, and the different (non-generic)
method that is usually preferred for the case of geometry.
In all cases, abstract types become fixed variables (or new constant symbols)
whose values are sets called interpreted
types (the respective ranges of variables of each type). For geometry, both
abstract types «Point» and «Straight line» become fixed
variables P and L, respectively designating the set
of all points, and the set of all straight lines.
The use of variable symbols will be left intact, taking values among some objects of set
theory (but not all).
While some objects of geometry, such as straight lines, are usually interpreted as
sets (of points), the generic method only uses pure elements as objects (or we may be ambiguous by calling them elements even if they are not pure).
The generic method will also convert structure symbols into fixed variables.
The interpretations of types and structure symbols (their values as fixed variables)
will determine the model, as they are its main components. The model,
which thus varies with these variables, is itself an object of set theory. This
integrates all theories we need as parts of the same set theory, while
gathering all their models inside a common model of set theory. This is
why models of set theory will be called universes.
FR : 1.3. Forme des
théories: notions, objets, méta-objets