1.7. Classes in set theory

The unified framework of theories

Attempts to formalize one-model theory in first-order logic cannot completely specify the meta-notions of «expressions» and «proofs», because as will be explained in 4.7 (Non-standard models of Arithmetic), descriptions of finite systems by first-order axioms cannot detect and exclude all non-standard ones, calling «non-standard system» an infinite system mistakenly (but consistently) described by a theory as «finite but indescribably large» (non-standard «proofs» are not actually valid). To fill this gap requires a second-order axiom, whose meaning is best expressed after insertion in set theory. As this insertion turns its components into free variables whose values define its model [T, M], their variability removes its main difference with model theory (the other difference is that model theory can also describe theories without models). This view of model theory as developed from set theory, which Parts 3 and 4 will almost achieve, will complete the grand tour of the foundations of mathematics after the formalization of set theory in the model theoretical framework.

Given a theory T so described, let T0 be the external theory, also inserted in set theory, which looks like a copy of T as any component k of T0 has a copy as an object serving as a component of T. Formally, T0 can be defined from T as made of the k such that («k» ∈ T) is true, where the notation «k» abbreviates a term of set theory designating k as an object, and the truth of this formula means that the value of this term in the universe belongs to T.

This picture forms a convenient unified framework for interpreting theories in models, encompassing both previous ones (set-theoretical and model-theoretical): all works on the theory T0 (expressions, proofs and other developments), have copies as objects formally described by the model theoretical development of set theory as the same «works on the theory T». Meanwhile in the same universe, any system M described as a «model of T» is indirectly also a set-theoretical model of T0. However, the power of this framework comes with a cost in legitimacy:

Classes, sets and meta-sets

For any theory, a class is a unary predicate A seen as the set of objects where A is true, that is «the class of all x such that A(x)». In particular for set theory, each set E is synonymous with the class of the x such that xE (defined by the formula xE with argument x and parameter E). However, this involves two different interpretations of the notion of set, that need to be distinguished as follows.

From now on, in the above unified framework, the theory T0 describing M and idealized as an object T, will be set theory itself. Taking it as an identical copy of the set theory serving as framework, amounts to taking the same set theory with two interpretations, that will be distinguished by giving the meta- prefix to the use of the framework interpretation.

Aside generic interpretations, set theory has a standard kind of interpretation into itself where each set is interpreted by the class (meta-set) of its elements, and each function is interpreted by the synonymous meta-function (more details in 1.D). This way, any set will be a class, while any class is a meta-set of objects. But some meta-sets of objects are not classes (no formula with parameters can define them); and some classes are not sets, such as the class of all sets (see Russell's paradox in 1.8), and the universe (class of all objects, defined by 1).

A kind of theoretical difference between both uses of set theory is irreducible (by the incompleteness theorem): for any given defined (invariant) axioms list for T, the existence of a model (universe), or equivalently its consistency, formalized as a set theoretical statement with the meta interpretation, cannot be logically deduced (a theorem) from the same axioms. This statement, or a fortiori that of the existence of a standard interpretation, thus forms an additional axiom of the set theory so used as framework.

Definiteness classes

Set theory accepts all objects as «elements» that can belong to sets and be operated by functions (to avoid endless further divisions between sets of elements, sets of sets, sets of functions, mixed sets...). This might be formalized keeping 3 types (elements, sets and functions), where each set would have a copy as element, identified by a functor from sets to elements, and the same for functions. But beyond these types, our set theory will anyway need more notions, which can only be conveniently formalized as classes. So, the notions of set and function will also be classes named by predicate symbols:

Set = «is a set»
Fnc = «is a function»

In first-order logic, any expression is ensured to take a definite value, for every data of a model and values of all free variables (by virtue of its syntactic correction, that is implicit in the concept of «expression»). But in set theory, this may still depend on the values of free variables.
So, an expression A (and any structure defined from it) will be called definite, if it actually takes a value for the given values of its free variables (arguments and parameters). This condition forms itself an everywhere definite predicate, expressed by a formula dA with the same free variables. Choosing one of these as argument, the class it defines is the meta-domain, called class of definiteness, of the unary structure defined by A.
Expressions should be only used where they are definite, which will be done rather naturally. The definiteness condition of (xE) is Set(E). That of the function evaluator f(x) is Fnc(f) ∧ x ∈ Dom f.
But the definiteness of the last formula needs a justification, given below.

Extended definiteness

A theory like set theory with partially definite structures can be formalized (translated) as a theory with one type and everywhere definite structures, keeping intact all expressions and their values wherever they are definite : models are translated one way by giving arbitrary values to indefinite structures (e.g. a constant value), and in the way back by ignoring those values. Thus, an expression with an indefinite subexpression may be declared definite if its final value does not depend on these extra values.

In particular, let us see AB and AB as definite if A is false (with respective values 0 and 1) even if B is not definite. That is, let us give them the same definiteness condition dA ∧ (A ⇒ dB) (breaking, for A ∧ B, the symmetry between A and B, that needs not be restored). This formula is itself made definite by the same rule provided that dA and dB were definite. This way, both formulas

A ∧ (BC)
(AB) ∧ C
have the same definiteness condition (dA ∧ (A ⇒ (dB ∧ (BdC)))).

Classes will be defined by everywhere definite predicates, which by the above rule can be easily expressed as follows.
Any predicate A can be extended beyond its domain of definiteness, in the form dAA (giving 0), or dAA (giving 1).
For any class A and any unary predicate B definite in all A, the class defined by AB, is called the subclass of A defined by B.

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. Formalizing types and structures
1.5. Expressions and definable structures
1.6. 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
Time in set theory
Interpretation of classes
Concepts of truth in mathematics
Philosophical aspects Time in model theory
Time between theories
2. Set theory (continued) 3. Algebra 4. Model theory
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FR : 1.7. Classes en théorie des ensembles