Spaces with non-trivial automorphisms will be interestingly seen as
non-unique in each isomorphic class (instead of picking one to represent them all),
for the following reason. Any isomorphism
*f* between 2 systems *E* and *F* induces a bijection
*g* ↦ *f*০*g* from Aut(*E*) to the set of isomorphisms
between *E* and *F*. Thus, a plurality of automorphisms
of *E* means a plurality of isomorphisms between *E*
and *F*. This plurality makes the difference between *E* and *F*
meaningful, as an object in *E* may correspond to several possible ones in
*F*, depending on the choice of isomorphism, so choosing an object in *E* does
not mean choosing an object in *F* in any invariant way. By contrast, the uniqueness of
isomorphism between standard models of ℕ or ℝ makes them unambiguously
play the role of each other (copies of each other), making their plurality superfluous.

For any set *E*, the relation of strong
preservation between its set Rel_{E} of relations and its set ⤹*E*
of permutations, defines a Galois connection
(Aut, sInv) similar to (End, Inv), where sInv
gives the set of *strong invariants* :

∀*P*⊂⤹*E*, sInv *P* = Inv (*P*
∪ -*P*) = Inv (*P*) ∩ Inv(-*P*) ⊂ Rel_{E}

∀*L*⊂Rel_{E}, ∀*P*⊂ ⤹*E*, *L* ⊂ sInv
*P* ⇔ *P* ⊂ Aut_{L} *E*.

Any relation defined by an expression with language

Constructions leave unchanged the automorphism groups seen as abstract groups, providing more types on which they act.

Structures

Both concepts of invariance can be still reconciled in different ways:

- For first-order logic, non-trivial automorphisms may externally exist over non-standard models of second-order arithmetic, thus also non-standard real closed fields (moving undefinable real numbers to infinitely close neighbors);
- In second-order logic, we may admit "second-order" ways of defining structures where any subset of ℕ is "defined" by the infinite data of all its elements(*).

Now starting with a mere concrete
category **C** of intended spaces whose class of morphisms is given by intuition,
let us review methods to produce invariant structures. We already gave the construction
of all invariant relations, elements of Inv(End *E*) or Inv(Aut *E*), from the
trajectories or orbits of tuples.

A language *L* ⊂ Inv *G* of first-order invariants may not suffice to
approach the concept of invariance defined by a given group :

- A well-describable
*L*independent of*E*may not fill Inv*G*(which is usually uncountable, but an uncountable*L*may be accepted either directly or as an additional type), in which case invariants outside*L*may be undefinable from*L*. - We need
*G*= Aut_{L}*E*, which will fail even for*L*= Inv*G*in the case of topology.

For example, Euclidean space geometry admits the type "plane", each plane is a set of points, and automorphisms can move any plane to any other plane. On Earth, "the sea level" names a plane, distinguishing its points from those above or below it, but Euclidean geometry does not see it as invariant, i.e. does not accept the name "sea level" in its language. A language accepting it, would form a different geometry.

(*)This bridge to Inv(Aut

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