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.

One-model theory

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.
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

Theory Kinds of objects (notions)
Generic theories Pure elements classified by types to play different roles
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 (operators, predicates), expressions (terms, formulas)...
Arithmetic Natural numbers
Linear Algebra Vectors, scalars...
Geometry Points, straight lines, circles...

Meta-objects

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, 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 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.

Set-theoretical interpretations

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...).
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.

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.

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.

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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