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 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
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 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 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.

Set-theoretical interpretation

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 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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