Philosophical complements to part 2

2.A. Time in set theory

Standard universes

The general idea is that the structures of U₁ (the considered interpretation of symbols of set theory in U₁ to make it a universe) must be the restrictions of those used for U₂. But it depends which structures we mean, and how we define their restrictions. From the property of preservation by restriction from U₂ to U₁ we shall specify for each version, more preservation properties on other symbols implicitly follow, thanks to the axioms of set theory in U₂. Any structure defined using only symbols with preserved meaning, is also preserved. The structures not preserved are anyway determined in U₁ as defined from those of U₂ with parameter U₁ (as a set or a unary predicate). This is why U₁ only needs to be specified as a meta-subset or class in U₂, keeping its structures implicit.

Let us focus on specifying what each version of standardness requires on sets (to which the diversity of versions is reducible, as the conditions on functions follow in the way essentially given by the fate of their graphs as sets, after naturally defining standardness for ordered pairs). The first structure for them is their class Set. The condition for this is that Set₁ = Set₂ ∩ U₁. Yet, arguing that Set is not really a structure but only a source of definiteness classes for effective structures, this condition may be weakened as Set₁ ⊂ Set₂ ∩ U₁ without essentially affecting the main 3 degrees of standardness described below.

The weak version (coming when seeing the role of sets as given by ∈) is

(ST) : ∈₁ = ∈₂ ∩ (U₁× Set₁).

The intermediate version (mentioned in 1.7: every set coincides with the meta-set of the same elements) is

(ST') : ∈₁ = ∈₂ ∩ (U₂× Set₁) ⊂ U₁× Set₁

The strongest version of standardness also requires the preservation of the powerset symbol ℘ (2.7):

(STP) : ℘₁ = ℘_2|Set1

(ST") : (ST) ∧ ∀E∈Set₁, ∀F∈Set₂, F ⊂₂ E ⇒ F ∈ Set₁

Ideally standard universes

(U₁ is standard in U₂) ⇔ (U₁ is standard in U₃).

A sub-universe U₁ of U₂ will be qualified as small in U₂ if it is known as a set there. This can be written U₁ ∈ Set₂ if U₂ is also itself (ST')-standard, otherwise ∃X∈Set₂, U₁ = ∈⃖₂(X).

To complete the concept of ideally standard universe, let us postulate that every two standard universes are small sub-universes of a third one.

This ideal is quite reasonable for (ST) and (ST'); but for (ST") it is somewhat more daring. As a first reason, it can be philosophically argued that any universe (standard or not) can be taken as a small sub-universe of another in the sense (ST'), but not always in the sense (ST") because seeing it as a set may provide means to define more subsets of given sets. In particular, any (ST")-standard universe being a small (ST") sub-universe of another, implies its fulfillment of the specification schema.

The realistic view of set theory

The above concepts provide a realistic meaning for set theory :

• Generic theories, seen axiomatically, are meant to be interpreted in a fixed model containing all values of variables and expressions;
• But set theory aims to "locally" interpret in the standard way any expression with fixed values of its free variables, independently of the choice of a standard universe, containing these values, where this interpretation may be processed.

Let us formalize some aspects of the plurality of standard universes which set theory aims to describe in its realistic view.

Standard multiverses (expansion of the set theoretical universe)

For any quasi-standard multiverse M, its union U = ⋃M forms another universe containing all members of M as small sub-universes. Then, for any expression with any values of its free variables in U there exists some U∈M containing these values and thus able to interpret the expression; this interpretation being independent of the chosen U, defines that of U. Therefore,

(M is a standard multiverse) ⇔ (the universe U is standard).

Can a set contain itself ?

• Take U\X as class of objects, and turn into urelements all sets E which contained a reflexive set (∃x∈E, x∈x), and all functions whose domain or image was such an E. (These sets cannot be rebuilt from the data of their elements, since each one has an element removed from the universe, namely itself ; any class of urelements can also be eliminated in the same way, turning other objects into urelements).
• Progressively rebuild a universe from scratch: this avoids all reflexive sets. Namely, from a given universe, first accept its urelements, then also accept, progressively along an endless "time", every set only made of previously accepted objects, and every function only involving such objects. Each set so accepted at some time, is formed as a collection of previously accepted objects. At the first time a set is so accepted as a collection of previously available objects, it could not be available yet as a possible element, thus it cannot be reflexive.
  As the reflexivity of a set is independent of context, a union of universes each devoid of reflexive sets, cannot contain one either.

• a universe with an urelement x and a non-reflexive set y such that x ∈ y,
  
• and a universe with a reflexive set x and an urelement y such that y ∉ x ?

The axiom of foundation, introduced in 5.3 (well-foundedness), will imply the absence of reflexive sets as a particular case ; its undecidability can be proven in ways essentially similar to the above arguments. But for most purposes, this axiom is as useless as the sets it excludes.