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Victor Pambuccian, Simple Axiom Systems for Euclidean Geometry, P SIMPLE AXIOM SYSTEMS FOR EUCLIDEAN GEOMETRY V ictor Pambuccian (Received July 1986) Wanda Szmielew in memoriam 1. Introduction Ever since Hilbert’s “Grundlagen der Geometrie” first appeared in 1899, a great deal of effort has been spent to simplify the axiom system (AS) for Euclidean geometry. W hat is the aim of these simplifications? That is, when do we stop simplifying? An ideal aim would be to ar­ rive at a most simple AS, but it is rather difficult to know what ‘most simple’ is supposed to mean. Some definitions for various concepts of (absolute) simplicity are proposed in [7]. A less ideal aim would be to arrive at a completely independent AS, i.e. one for which every axiom is independent of all the others. This aim is ‘less ideal’ because each ax­ iom, although not superfluous, may be further simplified (e.g. by taking only a special case of the given axiom). Completely independent AS’s for Euclidean geometry were proposed in [3], [6] and [12]. In this paper we give AS’s for Euclidean geometry, which are most simple according to a definition of simplicity proposed by G. Weaver [21]- These AS’s are based on significant results obtained by J.F. Rigby ([9], [10], [11]) and Wanda Szmielew [16], 2. Definitions and Notations We say that an ordered field F is Euclidean if Vx3y x > 0 —> x = y7. A Cartesian plane over F is the following structure C 2(F) = (F 2, Bp, D f ) where Bp = { (a, 6, c) € ( F 2)3 | € F 0<<<lAfc-a = <(c — a) and c / a } and D f = { (a,b, c, d) £ (F 2)4 | ||a — 6|| = ||c — <i|| and a / b }, with ||(x1,x 2)|| = y/x\ + x\. An AS for Cartesian planes over Euclidean fields is a set E of sentences in the language Lbd (first-order predicate calculus with two predicates: a ternary one lB ’ and a quaternary one ‘D ’ — standing for ‘Betweenness’ and ‘Equidistance’ respectively) for which M £ Mod(E) ifTM ~ C 2(F), Math. Chronicle 18 (1989), 63-74 for some Euclidean field F, where Mod(D) st ands for the class of models of E. Let (7n(£) =fy, where ('n(E) is t.hr* set of all logical consequences of E. Following Weaver [21], we say that a finite AS for a theory T issimple (and that T has simplicity degree m) if each of its axioms has no more than m variables and there is no AS for T all of whose axioms contain at most m — 1 variables. In the following section we give a simple AS for hereby proving that £'7 has simplicity degree 5. 3. The axiom system A l. Vabc B(abc) —*»B(cba). A2. Va&crf B(abd) A B(bcd) — B(abc). A3. Vabcd —>(c = d) A ( (B(abc) A B(abd)) V (B(abc) A B(dab)) V (B(bca) A B(bda) ) ) —*• B(acd) V B(cdn) V B(dac). A4. Va6c<f D(abcd) —*• ->(a = b) A -i(c = d). A5. Va6 ->(a = 6) —► D(abba). A6. (i) Vabcde D(abcd) A D(cdce) —► D(abce). (ii) Vaftcde D(abac) A D(ncde) —♦ D(abde). (iii) Vabcde D(abcd) A D(cdae) —► D(abae). A7. Vaftc 3rf Ve ->(a = 6) A -i(c = a) —► B(cad) A D(abad) A (/?(cae) A D(abae) —► d = e ). A8. Vabcde B(abc)/\(B(ade)\/B(aed)) AD(abad)hD(acae) —*• B(ade) A D(bcde). A9. Vabcde -«(c = rf)A D(acad) A D(bcbd) A B(abe) —► D(ecerf). A 10. Wabcde -<(b = d) A D(abad) A ( (B(abc) A B(adc)) V (/?(ca6) A B(ead) ) ) A D(acae) —► D(dcbe). A ll. Va6 3c -t(a = 6) —♦ /?(arfe) A D(cacb). A12. Vaftc 3rf ->((0 = 6) V (6 = c) V (c = a) V B(nbc) V B(bca) V B(cab)) -> D(dadb) A D(dbdc). A13. 3a6c -,((a = 6) V (6 = c) V (c = a) V B(abc) V B(bca) V B(cab)). A14. Vafccrfe -i(rf = e) A -*(a = 6) A ->(6 = c) A -«(c =a) A D(adac) A D(bdbc) A D(cdce) —► B(abc) V B(bca) V B(cab). A15. Vafrcrf 3e ->(rf = 6) A B(abc) —► B(dbe) A D (nrac). 64 The formula *B(a6c)’ may be read ‘b lies between a and c\ while ‘D(abcdy is read ‘a is as distant from b as c is from d\ or alternatively, ‘the segment ab is equal in length to the segment cd\ Here A 7 is a special case of the axiom of segment construction (which is A 7'; cf. [20]), first considered by J.F. Rigby ([9], [10], [11]); A9, A 10 are special cases of Tarski’s ‘five-segment axiom’ (which is A8'; cf. [20]); A8, A 10 are due to Rigby [11] (A8 is used in Proposition 2 in Book 1 of Euclid’s Elements, where it is derived from “Common notion 3”) and A9 to H.G. Forder [2]. A ll states that each segment has a midpoint, A 12 states that for any triangle the centre of its circumcircle exists; it is a form of Euclid’s parallel postulate; A 13 is a lower-dimension axiom, A 14 an upper-dimension axiom; A 15 is the ‘circle axiom’, stating that the circle drawn with radius greater than the distance from its centre to a given line intersects that line. Wanda Szmielew’s [16] AS for Cartesian planes over Euclidean fields consists of A1 - A2, A4 - A5, A ll - A 15 and A6'. Vabcdef D(cdab) A D(cdef) —► D(abef), AT. Vpacd 3! 6 ->(a = p) A -»(c =d) —♦ B(pab) A D(abcd), A8'. Vabca'b'c'pp' ->(a = b) A ->(p = c) A ~’(p/ = c') A D(abc) A #(a'6'c') AD(aba'b') A D{bcb'c') A D(pap'a') A D(pbp'b') —► D(pcp'c')} with the difference that she lets D(aabb) and B(aab) be true for all a and 6, therefore having Va6cD(a6cc) —+ a = b instead of A4, Va6 D(abba) instead of A5, Vpacd 36 B(pab) A D(abcd) instead of A7;; ~>(p = c) A ->(p' = c’) in A8' and -<(d = b) in A 15 are deleted. Obviously, this is a matter of taste and has no bearing on the rest of the AS. 4. Some proofs 1. Va6c B(abc) —* -<(a = c). Proof. Suppose B(aba). By A 13 3d -<(d = b). By A 15 3e B(dbe) A D(aeaa), contradicting A4. 2. Vafrc B(abc) —* ->(a = 6) A ->(6 = c). Proof. Suppose B(aac). Then B(aac) A B(aac) —► B(aaa) (A2). This contradicts (1). Suppose B(caa). Then B(aac), by A l. 3. Vabcd B(abc) A B(bad) —► ->(c = d). Proof. Suppose c = d. Then B(abc) A B(bac) —► B(aba) (A2), contra­ dicting (1). 65 4. Va6 3rdr -<(n = 6) —* -<(c = r/)A~>(c = r)A-'(d — r)AB(acb)AB(bad) A B(abr). Proof. 3c B(arb) A D(cacb) (A ll), 3c B(abc) A D(babe) (A7), B(abe) —*• B(eba), B(acb) —► B(bra) (A l), B(eba) A B(bca) —► B(ebr) (A2), B(ebc) —» ->(c = c) (1), 3rf Z?(6a<f) A D(«6arf) (A7), fl(6a(i) — (<ia6) (A l), Z?(<fa6) A fl(ac6) — fi(rfc6) (by A3, A2, A l), B(dcb) —* ->(<f = c) (2), B(a6e) A i?(6arf) — -.(c = rf) (3). Thus, by (4), any ‘line’ contains at least, 5 distinct points and */?’ satisfies A l, A2, A3, (1) and (2) so, by the results of [18], all the universal properties of the order relation on a line are satisfied by lB\ From now on this fact will ‘go without saying’. 5. Vo6 ->(a = b) —► D(abab). Proof. D(nbba) A D(baab) (A 5) D(abba) A D(baab) —»D(abab) (A6(iii)) 6. Vaftcrf B(nbc) A B(abd) A D(bcbd) —*■ c = d. Proof. Follows from (5) and A7 . 7. Va66' {B(abb') V B(ab'b) V (6 = 6')) A D(abab') ->6 = 6'. Proof. Suppose B(abb'). 3c B{b'ac) A D(ab'ac) (A7), B(b'ac) A B(abb') — /?(ca6), B(cab) A B(cab') A D(a6«6') - 6 = 6' (6). This contradicts B(abb'). The same contradiction follows if B(ab'b). Therefore 6 = 6'. 66 8. Va6c 3d —>(a = b)A~>(c = a) —► (B(adc)V B(acd)V(c = d)) AD(abad). Proof. 3e B(cae) A D(abae) (A7), 3d B(ead) A D(aead) (A7), B(cae) A B(ead) —► B(adc) V B(acd) V (c = <f), D(abae) A D(aea<f) —*■ D(abad) (A6(i)). 9. Va6c D(a6ac) —♦ D(acab). Proof. 36' (fl(afcfe') V £ (a i'6 ) V (6 = 6')) A Z)(aca6') (8), D(abac) A £>(aca6') —*■D(abab') (A6(i)), (B(abb') V fl(a&'6) V (6 = 6')) A Z)(a6a6') -^6 = 6' (7), D(acab') A (6 = 6') —► D(aca6). 10. 'iabcb'c' B(abc) A B(ab'c') A D(abab') A D(bcb'c') —► D(acac'). Proof. Z?(atc) A B(a6'c') —*■ ->(a = 6) A ~>(a = c'), 3d (fl(aftd) V £(a<i&) V (rf = 6)) AD{ac'ad) (8), D(abab') — D(a6'a6) (9), D(ac'ad) A D(ab'ab) A B(ab'c') A (B(abd) V B(adb) V (<f = 6)) — B(abd) A D(b'c'bd) (A8), D(bcb'c') A D(b'c'bd) — D(bcbd) (A6(iii)), B(a6c) A B(abd) A D(bcbd) -+c = d (6), D(ac'ad) A (c = cf) —► £>(ac'ac), D(ac'ac) —*■ D(acac') (9).
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