5.2. First-order invariants in concrete categories

Multiple automorphisms and models

Most geometries, except RCF, have a big group of automorphisms in each model. To produce such spaces from a framework without automorphisms (ZF with only sets and no pure elements, or second-order arithmetic), requires not only developments but also to forget some structures, and notice how some permutations then become automorphisms.

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 gfg 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.

The Galois connection (Aut, sInv)

For any set E, the relation of strong preservation between its set RelE 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) ⊂ RelE
L⊂RelE, ∀P⊂ ⤹E, L ⊂ sInv PP ⊂ AutL E.

For any subgroup G of ⤹E (automorphism group in any category), sInv G = Inv G.
Any relation defined by an expression with language LR without parameters is invariant by automorphisms, so is in Inv(AutL E). This means that adding it to L preserves AutL E (like other sets of isomorphisms between models).
Constructions leave unchanged the automorphism groups seen as abstract groups, providing more types on which they act.

The double meaning of "invariance"

Unlike provability which by the completeness theorem ranges over all truths of models, definability by language L without parameters may not fill Inv(AutL(E)), for the following reason.
Structures r ∈ Inv(AutL(E)), distinguishing any tuple tr from those outside r, are those for which no element of AutL(E) can move t out. This suggests that t is not similar to anyone outside r with respect to L, unless the trouble is to choose an automorphism witnessing the similarity. But this dissimilarity, and thus r itself, may be formally inexpressible from L as it may require infinitely complex descriptions. For example, there is no nontrivial automorphism in ℝ, or in a model of second-order arithmetic, yet not all objects of an uncountable type (real numbers or sets of integers) can be defined without parameter (but, as objects, they are definable with a parameter: themselves).
Both concepts of invariance can be still reconciled in different ways: A set of axioms of a theory can be seen as a mere tool to approach an intended range of models, on the other side of the (Mod,Tru) connection. Similarly, a language may be a mere tool to approach an intended concept of invariance better defined by a group G on the other side of the (Aut, sInv) connection. Specifying the geometry of a space E by a permutation group G of E intended as Aut(E), was a core idea of Felix Klein's Erlangen Program. Thus, "invariance" in geometry will be meant by G, and equivalently qualify structures defined with parameters in ℝ or among structures over ℝ only.

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 :

For this and more reasons which will appear later, the formalization of geometries needs second-order invariants.
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(AutL(E)) by means of extended kinds of "definitions" (with infinite amounts of data) has been investigated by Borner, Martin Goldstern, and Saharon Shelah : short presentation in .ps - full article.

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