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 is the choice of a
precise version of one-model theory, describing the admissible
forms of contents for T, what such contents mean about M,
and how their consequences can be deduced. Here are those we shall see,
roughly ordered from the poorest to the most expressive (though the order depends
on the ways to relate them):
We shall first describe the main two of them in parallel: first-order
logic, and set theory.
- First-order logic, describing what will be called here generic theories;
- Duality (important for geometry);
- Second-order logic;
- Higher-order logic;
- Set theory has its special framework
convertible into first-order logic.
All these frameworks except set theory, 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 (operators, predicates),
expressions (terms, formulas)...
||Points, straight 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 consists in
3 successive lists (kinds) of components,
where those of the latter
2 kinds are finite systems made from those of the previous kind(s).
A foundation of a theory is an initial choice of this content (describing
its intended models). Then, developments will
expand this content by additional components of each kind,
preserving its meaning (describing the essentially same models)
as will be explained later.
- A list of abstract types to serve as the names of types;
- A language (vocabulary): list of structure
symbols, names of the structures that will form an intended system by
relating objects with given types. Possible first-order structures (those allowed by first-order logic)
are predicates and operators (i.e. relations, operations and constants, see 1.4).
More powerful structures can come from set theoretical tools or as
packs of an additional type with first-order structures, as will be explained with second-order logic.
- A list of axioms chosen among ground formulas expressible in the
theory (see 1.5.), to request these formulas to be true when symbols
are interpreted in intended systems (models).
Any generic theory can be inserted (translated) into set theory by converting its
components into components of set theory. This is the usual
view of ordinary mathematics, studying many systems as «sets
with relations or operations such that...», with possible
connections between them. Let us present both the generic method
which works for any generic theory, and the different (non-generic)
method that is usually preferred for geometry.
In all cases, abstract types become names of sets, i.e.
fixed variables (or new constant symbols) whose values are sets called
interpreted types which will serve as ranges of variables of each type
(whose use is otherwise left intact).
The generic method converts all objects into pure elements,
but other methods called standard for specific theories (geometry,
or set theory in its standard interpretation) may do otherwise.
For example in geometry, both
abstract types «Point» and «Straight line» become fixed
variables P and L, respectively designating the set
of all points (which are pure elements), and the set of all straight lines (which
are sets of points).
The generic method will also convert structure symbols into fixed variables.
Possible interpretations of types and structure symbols (their values as fixed variables),
define a choice of model. Models become objects of set theory, owing their
multiplicity to the variation of these variables. This
integrates all needed theories as parts of the same set theory, while
gathering all their models inside a common model of set theory. This is
why a model of set theory will be called a universe.
Another procedure (sections 1.9 and 1.10, less useful to ordinary mathematics)
will convert the framework of set theory into first-order logic. As this does not reverse
the above translation, both frameworks of first-order logic and set theory remain
FR : 1.3. Forme des
théories: notions, objets, méta-objets