Axiom systems implying infinity in the foundations of geometry

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Axiom systems implying infinity in the foundations of geometry Victor Pambuccian1

· Rolf Struve2

Received: 22 August 2020 / Accepted: 11 September 2020 © The Managing Editors 2020

Abstract This is a survey of axiom systems for fragments of naturally encountered geometries which are just barely strong enough to imply that there are infinitely many objects in the universe of any of its models. Keywords Axiom system · Infinity · Ordered geometry · Metric planes · Hyperbolic geometry Mathematics Subject Classification 51F05 · 51G05 · 51M10

1 Introduction To reach the conclusion that there are infinitely many positive integers, one needs just two axioms, ¬S(x) = 0 and S(x) = S(y) → x = y, where we may think of S as the successor function. The question arises: How many axioms are needed in naturally occurring geometries, so that one can prove that there are infinitely many objects (points, lines, etc.)? Put differently, which fragments of geometry allow for the possibility of finite models, and when does the fragment become too rich to allow for finite models? Where does the junction between the finite and the infinite lie in the axiomatic foundation of geometry? We will explore the answers we find to this question in the literature on the axiomatic foundation of geometry. The geometries we will turn to are: the geometry of order, a rather austere plane absolute geometry, and geometries of hyperbolic type.

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Victor Pambuccian [email protected] Rolf Struve [email protected]

1

School of Mathematical and Natural Sciences, Arizona State University, West campus, Phoenix, AZ 85069-7100, USA

2

Bochum, Germany

123

Beitr Algebra Geom

2 The geometry of order 2.1 Dense or unending orders One way to express the geometry of order is by means of a ternary relation Z among individual variables to be interpreted as points, with Z (abc) to be read as ‘point b lies between a and c’ (in this variant of the predicate of betweenness, to be referred to as strict betweenness, point b must be different from a and from c, and the points a and c are supposed to be distinct). This is how Veblen (1904) expressed it. Previously, Hilbert (1977) used a language in which a variant of ordered geometry was proposed containing, besides Z , another sort of variables, for lines, as well as a point-line incidence relation. Collinearity can be easily defined in terms of Z alone, by L(abc) ⇔ Z (abc) ∨ Z (bca) ∨ Z (cab) ∨ a = b ∨ b = c ∨ c = a and we say that a point x is on line ab (with a = b) if L(abx) holds. The axiom systems {A1–A6} and {A1–A5, A7} suffice to prove that there are infinitely many points: (∃ab) a = b Z (abc) → Z (cba) Z (abc) → ¬Z (acb) Z (xab) ∧ Z (ayb) → Z (xay) (Z (abx) ∨ Z (bxa) ∨ Z (xab)) ∧ (Z (aby) ∨ Z (bya) ∨ Z (yab)) ∧ x = y → (Z (ax y) ∨ Z (x ya) ∨ Z (yax)) A 6 (∀ab)(∃x) [a = b → Z (axb)] A 7 (∀ab)(∃x) [a = b → Z (abx)]

A1 A2 A3 A4 A5

A1 states that there are two distinct points, A2 that betweenness is a symmetric relation in its endpoints, A3, in conjunction with A2, that no more than one point among three given

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