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
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 ↦ f০g 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
For any subgroup G of ⤹E (automorphism group in any category),
sInv G = Inv G.
∀L⊂RelE, ∀P⊂ ⤹E, L ⊂ sInv
P ⇔ P ⊂ AutL E.
Any relation defined
by an expression with language L⊂R 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 t∈r 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
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
Thus, "invariance" in geometry will be
meant by G, and equivalently qualify structures defined with
parameters in ℝ or among structures over ℝ only.
- 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
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
- 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 = AutL 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(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.
Set Theory and Foundations of Mathematics