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).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× Set1i.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|Set1This additional condition can also be written without explicit use of ℘ as
(ST") : (ST) ∧ ∀E∈Set1, ∀F∈Set2, F ⊂2 E ⇒ F ∈ Set1These conditions are related by (ST") ⇔ ((ST') ∧ (STP)) ⇒ (STF).
(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 above concepts provide a realistic meaning for set theory :
Let us formalize some aspects of the plurality of standard universes which set theory aims to describe in its realistic view.
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).But U cannot belong to M, because M has no greatest element. So, no single standard multiverse can ever contain all possible standard universes.
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|
Formalization of set theory |
2.2. Set generation principle
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