The 3-dimensional Euclidean geometry is the first and most obvious
theory of physics,
describing our physical space. To form a clear and complete mathematical
theory, this description must be idealized, accepting the simple properties
that seem to fit with ordinary physical measures, as if they were exactly
true, regardless the physical limitations of any attempt
at accuracy verifications. Any two points are regarded as
either equal or different, and between any two distinct points there
are other points and so on, which logically implies that any segment
contains infinitely many points. This infinity just comes as a good
approximation, as points very close to each other are physically harder
and harder to distinguish, with no clear distinction of any scale where
this distinction may become impossible or meaningless (it may
be argued to be the Planck length, far too small to be measured as
it is much smaller than nuclei of atoms).

Actually, this geometry is known to be inaccurate as a description of
physical space, since General Relativity theory is established as a more accurate
description of the physical space-time, which comes down to Euclidean
Geometry as a usually quite good
approximation (of its pure space aspect, part of Special
Relativity as a description of space-time).

While not exactly a physical theory, Euclidean plane geometry is useful for
educational purposes (being more directly visible and simpler than Euclidean
space geometry but with some complexity, thus a good intermediate step to
understand it), its relevance to aspects of physical space (describing flat
surfaces: screens, papers, pieces of land), and as an understanding of
complex numbers, which play key roles both for pure mathematics, and for
physics (especially the formalisms of waves and quantum physics).

- Types and
structures forming a weak
second-order theory :
- One base type called the "space" whose elements are the "points";
- Some types of structures over it: traditional ones are kinds of figures which are sets of points (straight lines, circles or spheres...) but other types will be needed.

- Axioms include a second-order one, of "continuity". Like the induction axiom of arithmetic, first-order logic can interpret it by an axiom schema restricting its range of application to sets of points definable with parameters. This forms the logically complete system of Tarski's first-order axioms of Euclidean plane geometry.

- a self-sufficient description of ℝ
- a description of all the rest by first-order logic using it, with finite lists of types, symbols and axioms which suffice insofar as ℝ is assumed to keep its standard interpretation.

ℝ can also be constructed from ℘(ℕ), using the decomposition of positive real numbers in integral part and fractional part, and representing the latter by its binary expansion as a subset of ℕ. Even the RCA

But the structures (0,1, +,⋅) of ℝ are weaker than those from that developed second-order arithmetic, as they do not suffice to distinguish integers (define either ℕ or ℤ) as a subset of ℝ: each standard number can be individually defined there but ℕ cannot be defined any better than the set of its standard elements ("

The diversity of non-standard real closed fields (non-isomorphic to ℝ) is illustrated though not exhausted by those constructed as above from non-standard models of second-order arithmetic. Namely, some contain non-standard numbers (larger than any standard natural number, so their inverses are infinitesimals), while others, having an (elementary) embedding into the standard ℝ, miss some undefinable numbers from it.

As RCF can only define numbers as zeros
of its polynomials (solutions of algebraic equations), whose existence is deduced from topological
completeness, the poorest real closed field is the set of real algebraic numbers (zeros
of polynomials with standard rational coefficients, while the zeros of polynomials with real
algebraic coefficients are among those of more complicated polynomials with rational coefficients).
Ironically for a foundation of geometry, it does not contain the number π (which is
not an algebraic number). In practice, physical applications involve some more
powerful framework, with tools of analysis
from RCA_{0} beyond RCF, such as trigonometric functions and
exponentials, including π.

The "framework" of the use of ℝ to formalize geometries, suffices to prove for "completely specified" geometries (specifying the dimension, which suffices for Euclidean geometry but not always otherwise):

- Semantic consistency: the existence of models (built using
ℝ
^{n}) - Semantic completeness : between any two models, isomorphisms can be defined with parameters (namely, choices of coordinate systems). First-order logic still sees them as incomplete, with isomorphism classes following those of RCF

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.

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.

In spaces of classical geometries with given finite dimensions, the endomorphism monoids and automorphism groups (which are second-order invariants) are first-order constructible, by a number of parameters which depends on the dimension, such as the image of one tuple with enough elements. For example, rotations (automorphisms in Euclidean geometry) can be specified by their center or axis, angle... but such a first-order constructibility no more holds for infinite dimensional spaces.

(*) in honor to the Greek geometer Euclid, who around 300 BC, published the

(**)This bridge to Inv(Aut

6. Geometry

Next : 6.2. Affine spaces

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