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.

One-model theory

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 formally describes T and its relation with M.
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 (listed in philosophical aspects) will be introduced later.
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

Theory Kinds of objects (notions)
Generic theory Pure elements classified by types
Set theory Elements, sets, functions, operations, relations, tuples...
Model theory Theories, systems and their components (listed next line)
  One-model theory     Objects, symbols, types, structures, expressions (terms, formulas)...
Arithmetic Natural numbers
Linear Algebra Vectors, scalars...
Geometry 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, expressions...

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:

Set-theoretical interpretation

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.

Set theory and Foundations of mathematics
1. First foundations of mathematics
1.1. Introduction to the foundations of mathematics
1.2. Variables, sets, functions and operations
1.3. Form of theories: notions, objects, meta-objects
1.4. Structures of mathematical systems
1.5. Expressions and definable structures
1.6. Logical connectives
1.7. Classes in set theory
1.8. Binders in set theory
1.9. Quantifiers
1.10. Formalization of set theory
1.11. Set generation principle
Philosophical aspects
Time in model theory
Time in set theory
Interpretation of classes
Concepts of truth in mathematics
2. Set theory (continued) 3. Algebra 4. Model theory
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FR : 1.3. Forme des théories: notions, objets, méta-objets