1.3. Form of theories: notions, objects and meta-objects
The variability of the model
In the way each theory describes its model, this model is
regarded as fixed, though in fact it is usually a mere «choice» of one model (interpretation) in
a wide (infinite) range of other existing, equally legitimate models of the same theory. This larger
view seeing a range of possible models, is that of model theory. Now this fixation of the model, like the
fixation of any variable, is but the abstract elementary act of picking any possibility, ignoring any
issue of how to specify an example in this range. Actually these «choice» and «existence» of models
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 of any consistent theory, but this construction is not really explicit (involving an infinity of
steps, where each step depends on an infinite knowledge). Regardless this difficulty of explicitation,
the attitude of implicitly fixing a model when working in any mathematical theory, remains the only
possible way of doing formalized mathematics, and must therefore be accepted.
Notions and objects
Each theory has its own list of notions, usually designated
by common names, formally serving as the kinds of variables it uses
; each model 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, and is usually
expressed using a different style of variable symbol for each. The
objects of a theory in a model, are all possible values
of its variables of all kinds (the elements of all its notions) in this model.
Any discussion of several theories T and systems M that may be
models of those T, takes place 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 focusing on one theory with a fixed model, the
variables T and M now fixed disappear from the list of variables.
Their kinds, the notions of theory and model, disappear from the notions list too.
This reduces the framework, from model theory, to that of
one-model theory. A model of one-model theory, is a system
[T,M] which combines a theory T with a model
M of T.
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
- Duality (for geometry) and the tensor formalism for linear algebra;
- 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 to play different roles
||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 interpretations of T's expressions there). But the same notions (even if from
a different logical framework) 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 lists of components of the following successive kinds,
where those of each of the latter
two kinds are finite systems made from those of the previous kind(s).
Developments of a theory will expand any such content by
additional components of each kind, preserving its meaning (describing
the essentially same models) beyond the case of theorems in ways that will
be explained in 4.8 and 4.9.
- 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. Those allowed by first-order logic,
thus called first-order structures,
are predicates and operators (i.e. relations, operations and constants, see 1.4). More powerful
structures called second-order
structures will be introduced later, coming from set theoretical tools or as
packs of an additional type with first-order structures.
- A list of axioms chosen among ground formulas expressible with this language
Any generic theory can be interpreted (inserted, translated) in the framework of 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 these systems. Let us introduce both the generic interpretations
applicable to any generic theory, and other ones usually preferred for particular theories.
All interpretations convert each abstract type into a symbol (name) designating a set
called interpreted type (serving as the range of variables of that type,
whose use is otherwise left intact). This symbol is usually a fixed variable in the generic case,
but can be accepted as constant symbol of set theory in special cases such as
numbers systems (ℕ, ℝ...).
Generic interpretations will also convert structure symbols into fixed variables, while
standard ones may define these from the language of set theory instead.
Any fixed values of all types and structure symbols, define a choice of model.
Models become objects of set theory, owing their
multiplicity to the variability of types and structure symbols. 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 is called a universe.
In generic interpretations, all objects (elements of interpreted types) are
pure elements, but other kinds of interpretations called standard
by convention for specific theories may do otherwise.
For example, standard interpretations of geometry represent points by
pure elements, but represent straight lines by sets of points.
Another procedure (sections 1.9 and 1.10, less useful to ordinary mathematics)
will convert the framework of set theory into first-order logic. But this is not the inverse
of the above conversion from generic theories into set theory, so that
both frameworks of first-order logic and set theory remain fundamentally different.
FR : 1.3. Forme des
théories: notions, objets, méta-objets