# Philosophical complements to part 2

## 2.A. Time in set theory

### Standard universes

Given two universes U1, U2, with U1U2, we shall define 3 distinct versions for the concept of U1 being standard in U2, also called a sub-universe of U2. We first informally introduced such a concept in 1.7 as the preservation of the roles of sets and functions; then in 1.B with the other meaning (STF) of standardness of ℕ, i.e. the interpretation of finiteness (applicable to any universe as mentioned in 1.D).

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 = Set2U1. 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 ⊂ Set2U1 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).

Actually, this condition will suffice to imply (STF) once adopted the axiom of foundation for U2 and further reasonable conditions which will be detailed later. Otherwise we need to add it as another requirement in the concept of standardness, in the form ℕ1 = ℕ2 (if the axiom of infinity is adopted in both) or anyway in terms of the class Fin of finite sets as Fin1 ⊂ Fin2 which is equivalent to Fin1 = Fin2 ∩ Set1 (for reasons a bit subtle).

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

i.e. every set in U1 is interpreted by U2 as included in U1. This implies the preservation of bounded quantifiers (if the sub-formula keeps its values, namely does not involve ℘) and of any operator giving a set determined by the data of its elements, namely the operators given by the set generation principle. Special constructions exist (such as those of internal set theory) of sub-universes not fitting this condition and still preserving bounded quantifiers on formulas with parameters in U1; but the equivalence holds with the faithfulness of bounded quantifiers on formulas with parameters in U2 (as we shall see when justifying the set generation principle in 2.B).

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

(STP) : ℘1 = ℘2|Set1

This additional condition can also be written without explicit use of ℘ as

(ST") : (ST) ∧ ∀E∈Set1, ∀F∈Set2, F2 EF ∈ Set1

These conditions are related by (ST") ⇔ ((ST') ∧ (STP)) ⇒ (STF).
Indeed, (STF) comes by the fact finiteness is definable from ℘ (independently of any additional assumption which was needed with the mere (ST)) ; this can also be seen considering that arithmetic is specified as a second-order theory, which can be faithfully interpreted by set theory with ℘.
The independence of the axiom of choice (further commented in 5.5) and other statements involving ℘, can be roughly explained by the possibility to construct universes which do not preserve the essential meaning of ℘. Following convenience, one such construction happens to fit (ST'); others may fit (ST) ∧ (STP) but not (ST') while (ST')-standard copies of their results also exist.

### Ideally standard universes

For any 3 universes U1U2U3 where U2 is standard in U3,

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

Thus the idea to consider the standardness of U1 as an absolute property, independent of the external universe in which it is described... provided that this external universe is itself standard. This does not define standardness as an absolute concept (which cannot be defined anyway), but suggests that such a concept may make ideal sense.

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

From now on we shall assume a fixed version of "standardness" meant as either (ST') with a set theory without powerset, or (ST") with a set theory with powerset. In either case, all bounded expressions keep the same values from a standard universe to another, thus called their standard values.

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.
In the next section (2.B) the meaningfulness of the (ST") ideal will appear philosophically equivalent to the realism of admitting ℘ in set theory.

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)

Let us call quasi-standard multiverse (resp. standard multiverse) any collection (range) of universes (resp. standard universes), where any 2 of them are small sub-universes of a third one. Let us require it to also contain all sub-universes of its members, a quality easy to fulfill, taking a multiverse where it did not hold and adding all these sub-universes to it. We shall say that the set theoretical universe expands when it ranges over a standard multiverse.

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 UM 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).

But U cannot belong to M, because M has no greatest element. So, no single standard multiverse can ever contain all possible standard universes.
For a (ST")-quasi-standard multiverse, keeping in U the same powerset of a given set as in member universes, may lead to the failure of the specification schema in U. Here comes another side of the postulate attached to the ideal (ST")-standardness : that with (ST")-standard multiverses, this failure never happens.

### Can a set contain itself ?

Let us call a set reflexive if it contains itself.
The proof of Russell's paradox [1.8], shows that the class of non-reflexive sets (Set(x) ∧ xx) cannot be a set (as it is not included in any set). So, many sets are non-reflexive ; but can reflexive sets exist ? this is undecidable; here is why.

From any universe U with a nonempty class X of reflexive sets, another universe containing none can be made by 2 different methods
• Take U\X as class of objects, and turn into urelements all sets E which contained a reflexive set (∃xE, xx), 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.
On the other hand, universes with reflexive sets can be produced as follows:

Riddle. What is the difference between
• a universe with an urelement x and a non-reflexive set y such that xy,
• and a universe with a reflexive set x and an urelement y such that yx ?
Answer (click to show) : The role of the set containing x but not y, played by y in the former universe, is played by x in the latter.

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.

 Set theory and Foundations of Mathematics 1. First foundations of mathematics 2. Set theory 2.1. Formalization of set theory 2.2. Set generation principle 2.3. Tuples 2.4. Uniqueness quantifiers 2.5. Families, Boolean operators on sets 2.6. Products, graphs and composition 2.7. The powerset 2.8. Injections, bijections 2.9. Properties of binary relations 2.10. Axiom of choice ⇦ 2.A. Time in set theory⇨ 2.B. Interpretation of classes 2.C. Concepts of truth in mathematics 3. Algebra ⇨ 3.1. Galois connection 4. Arithmetic - 5. Second-order foundations

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FR : Temps en théorie des ensembles