Introduction to the foundations of geometry

This text is under construction.

What is geometry

Geometry is the family name of a fuzzy range of mathematical theories, some of whom have useful relevance to describe the space of our physical universe (to describe possible figures there, and relate their diverse measures), and other aspects of the laws of physics. Diverse geometries can be explored for the sake of pure mathematics (as studies of abstract realities), disregarding their possible connections to physics, and this is even needed as a theoretical background to better understand those relevant to physics as particular cases.
The first two geometries classically considered, were those of “the plane” and “the space” as they naturally appear. They are now called Euclidean geometries(*), distinguished by their dimension, respectively 2 (for the plane geometry), and 3 (space geometry).

The 3-dimensional Euclidean geometry is the first and most obvious theory of physics, describing our physical space. To be a clear (simple) 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. For example, it must regard any two points as either equal, or different, and that between any two different points there are other points and so on, which logically implies that any segment contains infinitely many points. This infinity so comes as a mere good approximation, as intermediates between points very close to each other are physically harder and harder to distinguish, with no clear way to figure out at which scale may this distinction become impossible or meaningless (this may be argued to be the Planck length, which is far too small to be measured anyway, 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).

Usually, geometries are weak second-order theories with one base type called the "space" whose elements are the "points", seen as pure elements. Traditionally considered other types are kinds of figures which are sets of points (straight lines, circles or spheres...) but types beyond sets of points will be needed for diverse reasons.

Multiple automorphisms and models

Most geometries can be characterized as having a big group of automorphisms in each model. Such theories are not directly obtained by developing theories without automorphisms (ZF with only sets and no pure elements, or second-order arithmetic), but obtaining them from there requires to also forget some structures, and notice how some permutations then become automorphisms.

By its general definition, the automorphism group for any theory with a finite language is a second-order construction. In classical geometries (but not topology), it is even first-order constructible from the space. For example, rotations are automorphisms in Euclidean geometries, and can be specified by just a few parameters (depending on the dimension: center or axis, angle... or the image of one tuple). Anyway geometries generally admit their own automorphism group either as a type, or as a subset of a type (especially the type of endomorphisms, if different).

Like in a set theoretical framework, for geometries with nontrivial automorphisms, multiple models (spaces) may be interestingly considered in each isomorphic class (instead of taking 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, a unique isomorphism between models makes them unambiguously play the role of each other (copies of each other), rendering their plurality superfluous.

The Galois connection (Aut, sInv) and the double meaning of "invariance"

For any set E, the relation of strong preservation between RelE and the symmetric groupE induces a Galois connection (sInv, Aut) between their powersets (similar to (Inv, End) previously seen): for any set G⊂ ⤹E of permutations, the set of strong invariants of G is

sInv (G) = Inv (G ∪ -G) = Inv (G) ∩ Inv(-G)
L⊂RelE, ∀G⊂ ⤹E, L⊂sInv (G) ⇔ G⊂AutL(E).

If G is a group then G = -G, thus sInv (G) = Inv (G). As we know, the set Inv (G) of invariants of G is made of orbits of tuples, and any unions of these.

We initially conceived invariant structures as those defined by expressions without parameters. This implies invariance by automorphisms. Like proofs explore truths (theorems), that is the closure of the set of axioms, whose addition to axioms preserves the class of models, definitions from a language LR explore its closure sInv(AutL(E)) = Inv(AutL(E)) ⊃ L, whose addition to L leaves unchanged the set of automorphisms of each model (among other isomorphisms between models). Constructions leave unchanged the groups of automorphisms as abstract groups, just providing more types on which these groups act.
Our last scheme of constructions (set of all structures defined by an expression with all values of parameters), was that of an invariant second-order structure that is a set of non-invariant first-order structures. Well-defined by that formula, its invariance by automorphisms means that automorphisms preserve that set while acting as transformations on it.

While the completeness theorem gave a converse for the similar concept of provability, such a converse no more strictly holds here. Let us shortly explain this subtle difference between both concepts of invariance, before undertaking to confuse them in a way which will fit for most needs of studying geometry at a basic level.
Two tuples can be distinguished by a structure r ∈ Inv(AutL(E)), when they cannot be moved one to the other by an element of AutL(E). Intuitively, it suggests that these tuples are not similar with respect to L, unless there is a trouble to choose an automorphism witnessing their similarity. One might then hope for this dissimilarity, and thus r itself, to be expressible from structures in L. But this may require infinitely complex definitions, inexpressible with finite formulas. For example, there is no nontrivial automorphism in the system of real numbers (ℝ, 0,1,+,⋅), or in a model of second-order arithmetic, yet objects of an uncountable type (real numbers or sets of integers) cannot be all defined without parameter (but, as objects, they are definable with a parameter: themselves).
There are still 2 viewpoints which may make both concepts of invariance somehow coincide : Now starting from a permutation group G intended as the set of automorphism of a space E, can we rebuild its geometry, looking for a good list L of a few structures from Inv G, which would suffice to define other useful invariant structures (not the whole set of them, which is uncountable), at least so that G=AutL ? This was a core idea of Felix Klein's Erlangen Program for the foundations of geometry.

Completeness of first-order geometry

The difficulties in choosing a logical framework for geometries, turn out to be reducible to a specific case: the theory of the set ℝ of real numbers. Indeed, while formalizations of geometries without numbers are possible, the type ℝ can anyway be restored from them by a construction. Then they can be conveniently formalized using ℝ as a base type aside the space, beyond which all the rest can be formalized in first-order logic, with finite lists of types, symbols and axioms which suffice insofar as ℝ is assumed to keep its standard interpretation. So, this way of "using ℝ with its standard interpretation" may be regarded as all we need in guise of framework for geometry beyond first-order logic.
The system of real numbers is essentially unique in the sense that This "framework" (use of real numbers) suffices to prove that diverse geometries have existing models of (by constructing a type of "points", as tuples of real numbers or more complicated things... and forgetting some structures) and are semantically complete: that between any two models of any given geometry with the same dimension there is an isomorphism (definable with parameters, typically by choosing coordinate systems and identifying points with the same tuple of values of coordinates). For first-order logic, the isomorphism classes of models of each complete geometry follow the isomorphism classes of real closed fields. Such fields can be non-standard (non-isomorphic to ℝ) either because they have nonstandard numbers (larger than any standard natural number) or because some "finite" but undefinable numbers are missing.

Geometries may be interestingly generalized by replacing ℝ by something else such as the set ℂ of complex numbers, still keeping many things work the same.

ℝ can be built from P(ℕ), using binary expansions. Its non-standard models either contain non-standard numbers, or fail to contain some undefinable numbers. As the first-order theory of real closed fields only has algebraic equations as a means to define numbers, the poorest model is just made of algebraic numbers.

We get an expression of any geometry as a complete first-order axiomatic theory, by means of its relation with ℝ together with such an axiomatization of ℝ.
This quality of completeness comes in contrast with the incompleteness theorem of arithmetic. It avoids contradiction with it, by the fact that formulas of arithmetic (and especially the undecidable ones) cannot be expressed as formulas in the first-order theory of real numbers. Because this theory cannot express the predicate over real numbers "to be an integer", as the formula "x=0 or x=1 or x=2 or...(and so on to infinity)", well, is not a formula.

The theory of real closed field obliges its models (the real closed fields) to contain all algebraic numbers (solutions of algebraic equations with (standard) rational or integral coefficients).  However, it cannot express the existence of any number that is not algebraic (because any finite list of first-order properties satisfied by a non-algebraic number, is also satisfied by some algebraic number).
In particular, such a theory cannot express the number π (which is not an algebraic number). What a pity for a geometry !

At first, we shall assume geometrical spaces and real numbers as known from experience (from secondary-level mathematics). Fully rigorous (set theoretical and axiomatic) foundations will be presented later.

Dimensions

The dimension n of a space can be intuitively understood as the arity of tuples of real numbers (called coordinates), needed to give the position of a point. In other words, the space has a bijective and continuous correspondance to ℝn (this correspondence is called a coordinates system). For each natural number n (say, nonzero to make something non-trivial) there is one n-dimensional Euclidean geometry, but also diverse other n-dimensional geometries, with different concepts and properties. Even infinite-dimensional geometries may be considered.
As some geometries (such as the spherical geometry) have no good global coordinates system, at least we put this as a local requirement : the existence of correspondences of its small enough regions with those of ℝn.
By the action of the automorphism group, the space of points usually has only one orbit (but vector spaces have 2 orbits, one being a singleton), so that when regarding the group of automorphisms as a space itself, it has even higher dimension.
This happens for many finite-dimensional spaces of interest, but yet not all:


(*) in honor to the Greek geometer Euclid, who around 300 BC, published the Elements, mathematical treaty gathering and methodologically ordering, with logic, axioms and proofs, the main mathematical knowledge of that time, with a special focus on plane and space geometries. It remained the most famous mathematical work until the 19th century, when its axioms and proofs were found incomplete. Only in 1899 a really complete axiomatic expression of Euclidean plane geometry was published by Hilbert.
(**)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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