Set theory as a unified framework

Structure definers in diverse theories

Let us call structure definer any binder B which, used on diverse expressions A, faithfully records the unary structure defined by A on some range E (type or class defined by an argument here implicit), i.e. its result S = (Bx, A(x)) can restore this structure by an evaluator V (symbol or expression) : V(S, x) = A(x) for all x in E. Admitting the use of negation and the possibility to interpret Booleans by objects (in a range with at least 2 objects, which is often the case), Russell's paradox shows that adding both following requirements on a structure definer in a theory would lead to contradiction :
  1. All such S belong to E
  2. V can occur in the expression A and use x anyhow in its arguments, namely V(x, x) is allowed, which makes sense as 1. ensures the definiteness of any V(S, S).
Let us list the remaining options. Set theory rejects 1. but keeps 2. But since 1. is rejected, keeping 2. may be or not be an issue depending on further details.

As will be explained in 4.9, extending a generic theory (whose ranges of binders were the types) by a new type K given as the set of all structures defined by a fixed expression A for all combinations of values of its parameters, forms a legitimate development of the theory (a construction). Indeed a binder on a variable structure symbol S with such a range K abbreviates a successive use of binders on all the parameters of A which replaces S. Here A and the system interpreting it come first, then the range K of the resulting S and their interpretations by V are created outside them : A has no sub-term with type K, thus does not use V (which has an argument of type K).

The notion of structure in first-order logic (as a one-model theory) has this similarity with the notion of set in set theory : for each given symbol type beyond constants, the class of all structures of that type is usually not a set, calling "sets" such ranges K (of structures defined by a fixed expression with variable parameters), or subclasses of these.
The fully developed theory with the infinity of such new types constructed for all possible expressions A, can become similar to set theory by gathering to a single type U all constructed types K of variable structures of the same symbol type (structures over the same sets), interpreted by the same symbol V (which could be already used by A). This merely packs into V different structures without conflict since they come from different types K of the first argument. This remains innocent (re-writing what can be done without it) as long as in the new theory, the binders of type U stay restricted to one of these "sets" K (or covered by finitely many of them, which are actually included in another one).

In set theory, the ranges of binders are the sets. Thus, beyond the simplifying advantage of removing types, set theory will get more power when accepting more classes as sets.

Other theories, which we shall ignore in the rest of this work, follow more daring options:

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». Indeed as will be explained in 4.7 (Non-standard models of Arithmetic), any first-order theory aiming to describe finite systems without size limit (such as expressions and proofs) inside its model (as classes included in a type), will still admit in some models some pseudo-finite ones, which are infinite systems it mistakes as «finite» though sees them larger than any size it can describe (as the latter is an infinity of properties which it cannot express as a whole to detect the contradiction ; these systems will also be called non-standard as «truly finite» will be the particular meaning of «standard» when qualifying kinds of systems which should normally be finite).
To fill this gap will require a second-order universal quantifier (1.10), whose meaning is best expressed (in appearance though not really completely) after insertion in set theory (whose concept of finiteness will be defined in 4.5). 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, will be exposed in Parts 3 and 4, completing the grand tour of the foundations of mathematics after the formalization of set theory in a logical 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. In some proper formalization, 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 forms a convenient unified framework for describing theories interpreted in models, encompassing both previous ones (set-theoretical and model-theoretical): all works of the theory T0 (expressions, proofs and other developments), have copies as objects formally described by the model theoretical development of set theory as works of the theory T. In the same universe, any system M described as a model of T is indirectly also a (set-theoretical) model of T0.
This powerful framework is bound to the following limits : So understood, the conditions of use of this unified framework of theories, are usually accepted as legitimate assumptions, by focusing on well-described theories (though no well-described set theory can be the "ultimate" one as mentioned below), interpreted in standard universes whose existence is admitted on philosophical grounds; this will be further discussed in philosophical pages.

Set theory as a unified framework of itself

The above unified framework is applicable to the case of set theory itself, thus expanding the tools of interpretation of set theory into itself already mentioned in 1.7. Namely, 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 interpreted by two universes.

A kind of theoretical difference between both uses of set theory will turn out to be irreducible (by the incompleteness theorem): for any given (invariant) formalization of set theory, the existence of a model of it (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, and thus also the stronger statement of the existence of a standard universe, thus forms an additional axiom of the set theory so used as framework.

Zeno's Paradox

Achilles runs after a turtle; whenever he crosses the distance to it, the turtle takes a new length ahead.
Seen from a height, a vehicle gone on a horizontal road approaches the horizon.
Particles are sent in accelerators closer and closer to the speed of light.
Can they reach their ends ?

Each example can be seen in two ways:

In each example, a physical measure of the «cost» to approach and eventually reach the targeted end, decides its «true» interpretation, according to whether this cost would be finite or infinite, which may differ from the first guess of a naive observer.
But the world of mathematics, free from all physical costs and where objects only play conventional roles, can accept both interpretations.

Each generic theory is «closed», as it can see its model (the ranges of its variables) as a whole (that is a set in its set theoretical formulation): by its use of binders over types (or classes), it «reaches the end» of its model, and thus sees it as «closed». But any possible framework for it (one-model theory and/or set theory) escapes this whole.
As explained in 1.7, set theory has multiple possible models : from the study of a given universe of sets, we can switch to that of a larger one with more sets (that we called meta-sets), and new functions between the new sets.

As this can be repeated endlessly, we need an «open» theory integrating each universe described by a theory, as a part (past) of a later universe, forming an endless sequence of growing realities, with no view of any definite totality. This role of an open theory will be played by set theory itself, with the way its expressions only bind variables on sets (1.8).