The general idea is that the structures of U_{1} (the considered interpretation of symbols of set theory in U_{1} to make it a universe) must be the restrictions of those used for U_{2}. But it depends which structures we mean, and how we define their restrictions. From the property of preservation by restriction from U_{2} to U_{1} we shall specify for each version, more preservation properties on other symbols implicitly follow, thanks to the axioms of set theory in U_{2}. Any structure defined using only symbols with preserved meaning, is also preserved. The structures not preserved are anyway determined in U_{1} as defined from those of U_{2} with parameter U_{1} (as a set or a unary predicate). This is why U_{1} only needs to be specified as a metasubset or class in U_{2}, 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 Set_{1} = Set_{2} ∩ U_{1}. 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 Set_{1} ⊂ Set_{2} ∩ U_{1} 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} ∩ (U_{1}× Set_{1}).
Actually, this condition will suffice to imply (STF) once adopted the axiom of foundation for U_{2} 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 ℕ_{1} = ℕ_{2} (if the axiom of infinity is adopted in both) or anyway in terms of the class Fin of finite sets as Fin_{1} ⊂ Fin_{2} which is equivalent to Fin_{1} = Fin_{2} ∩ Set_{1} (for reasons a bit subtle).The intermediate version (mentioned in 1.7: every set coincides with the metaset of the same elements) is
(ST') : ∈_{1} = ∈_{2} ∩ (U_{2}× Set_{1}) ⊂ U_{1}× Set_{1}
i.e. every set in U_{1} is interpreted by U_{2} as included in U_{1}. This implies the preservation of bounded quantifiers (if the subformula 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 subuniverses not fitting this condition and still preserving bounded quantifiers on formulas with parameters in U_{1}; but the equivalence holds with the faithfulness of bounded quantifiers on formulas with parameters in U_{2} (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} = ℘_{2Set1}
This additional condition can also be written without explicit use of ℘ as(ST") : (ST) ∧ ∀E∈Set_{1}, ∀F∈Set_{2}, F ⊂_{2} E ⇒ F ∈ Set_{1}
These conditions are related by (ST") ⇔ ((ST') ∧ (STP)) ⇒ (STF).(U_{1} is standard in U_{2}) ⇔ (U_{1} is standard in U_{3}).
Thus the idea to consider the standardness of U_{1} 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 subuniverse U_{1} of U_{2} will be qualified as small in U_{2} if it is known as a set there. This can be written U_{1} ∈ Set_{2} if U_{2} is also itself (ST')standard, otherwise ∃X∈Set_{2}, U_{1} = ∈⃖_{2}(X).
To complete the concept of ideally standard universe, let us postulate that every two standard universes are small subuniverses 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 subuniverse 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") subuniverse 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 quasistandard multiverse M, its union U = ⋃M forms another universe containing all members of M as small subuniverses. 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 (wellfoundedness), 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. Secondorder foundations 