Introduction to the foundations of geometry

What is geometry

For pure mathematics, 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), ignoring their possible connections to physics; and this is even needed as a theoretical background to better understand the cases relevant to physics. The first two classically considered geometries were those of “the plane” and “the space” as they seem. They are now called Euclidean geometries(*), respectively with dimension 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 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).

Usual geometries are second-order theories made of

The use of real numbers

The use of second-order axioms for geometries by which they escape first-order logic, turns out to be reducible to a specific case: the theory of the system ℝ of real numbers, that is an algebra whose language is made of constants 0,1 and binary operations +,⋅, from which subtraction and order can be defined : xy ⇔ (∃z, y = x + zz). While geometries admit formalizations without it, the type ℝ can anyway be restored from them by a construction. Geometries can be conveniently formalized in two parts : This "use of ℝ with its standard interpretation" sums up all the needed framework for geometry beyond first-order logic. Some geometries may be interestingly generalized, keeping many similar properties, by replacing ℝ by another system of numbers (especially the set ℂ of complex numbers).

In second-order logic, ℝ has a semantically complete description, whose second-order axiom (of continuity) describes its order as topologically complete. The reduction (restriction) of the axiom of topological completeness as an axiom schema of first-order logic, gives the logically complete first-order theory of real closed fields (RCF).
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 RCA0 subsystem of second-order arithmetic suffices to ensure the needed properties for this first-order development of ℝ out of ℘(ℕ), whose standardness is equivalent to the standardness of the resulting ℝ.
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 ("x∈ℕ", defined as "x=0 ∨ x=1 ∨ x=2 ∨...and so on to infinity", is not a formula). This weakness explains why RCF can be logically complete without contradicting the incompleteness theorem of arithmetic: RCF cannot express the undecidable formulas of arithmetic.

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 RCA0 beyond RCF, such as trigonometric functions and exponentials, including π.

Dimensions and semantic completeness

The dimension n of a space can be intuitively understood as the arity of tuples of real numbers needed to give the position of a point (called its coordinates), according to some map which is a continuous correspondence of the space to ℝn, called a coordinates system (it will be bijective in our first considered geometries but may be non-bijective in others). 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.
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):
(*) 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.

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