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<< 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 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 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 = 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 11. 'iabcdefo D(oaoc) A D(ocoe) A D(abcd) A D(cdef) —* D(abe/). Proof. 36' B(oab') A D(abab') (A7), 3d' B(ocd') A D(cdcd') (A7), 3 /' B(oef') A D(efef') (A7), D(abab') — D(ab'ab) (9), D(ab'ab) A D(abcd) —► D(ab'cd) (A6(ii)), 67 D{ab'rd) A D(cdcd') — D(ab'cd') (AO(i)), D(cdcd') -» D(cd'cd) (9), D(crfW) A D(cdr/) — D(rd'ef) (AO(ii)), D(cd'e f) A D(e f e/ ') — D(cd'rf’) (A6(i)), Z?(oa6') A B{ocd!) A D(oaoc) A D(ab'cd') —> D(ob'od') (10), D{ocd') A £(or/') A D(oro^) AD{cd'ef) — D(orf'o/') (10), D(oaoc) A D(ocoe) —► D(oaor) (A6(i)), D(ob'od') A D{od'of) - D(ob'of') (A6(i)), fl(oaft') A B(oef') A D(oaoe) A D(ob'of') — D(a6'c/') (A8), D(abab') A D{ab'ef') - D(a6e/') (A6(ii)), D {e fef) - D (c /'c /) (9), D(abef') A D {e fe f) - D(a6c/) (A6(i)). 12. Vabcdef D(abcd) A D(cdef) —*■ D(abef). Proof. If a, c, e are three different, not collinear points, i.e. if -•(B(ace)V Z?(cea)V/?(eac)V(a = c)V(c = f)V(e = a)), then, by A12, 3o £>(oaoc)A D(ocoe) so, by (11), (12) is true. Suppose a,c,e are collinear or are not different. If a = c or c = e or e = a then (12) is just A6(ii) or A6(i) or A6(iii) respectively. Suppose now B(nce) V B(cea) V B(eac). 3x ~>((x — c) V {x = c) V B(cxr) V B(xec) V B(ecx)) (A13), 3 /' B{xef')/\D{efef') (A7), D(cfef') - D(e/'e/) (9), D(cdef) A D(e/e/') — D(crfe /') (A6(i)) . Since e ,e ,/' as well as a ,e ,f as well as a ,c ,/' are three different, not collinear points, we have D(cdef') A D (c ff'e ) — D{cdf'e), D(abcd) A D(cdf'e) —► D(abf'e) and D(abf'e) A D{f'eef') — D(abef'). D(abef’) A D{ef'ef) - D{abef) (A6(i)). Since, with the aid of (12), we can prove (cf. [11], page 18) that D(abcd) —► D(cdab), AG' is proved. From now on, the methods of [9] and [11] can be used to prove A7' (cf. [11], pages 17, 18) and A8' (cf. [9], page 180), so all the axioms of [16] are consequences of our AS. This means that A1 - A 15 is an AS for 68 5. Simplicity Let T = Cn({ Let M = (It2, D m , where Dm = {a,b,c,d ( ) € (R 2)4 | a ^ 6, ||a - 6|| = ||c -d\\ }, LM = { (a,b,c,) £ (R 2)3 | a ^ c, 6 = ta + (1 — t)c, for some t € R\{0,1} } and Mm = {(a,6,c) 6 (R 2)3 | a ^ c, b = (l/2 )(a + c)}. T\ = 77i£,DLMM may be regarded as an ^BD-lheory (denoted T/) by substitutingB(abc) V B(bca)V B(cab) and B(abc) A D(babc) respectively for any occurrence of L(aftc) and M(abc) in a sentence in T\. It is easily checked that T C Cn(T{ U S) where S = {A l, A2, (1), (2), F I, F2, F3, F4}, F I. Va6c (B(abc) V B(acb) V (6 = c)) A D(abac) —*■ b = c, F2. Va6c 3d -i(a = b) A -«(c = a) —♦B(cad) A D(abad), F3. Va6c 3d -<(a = 6) A -*(c = a) —► B(acd) A D(abcd), F4. Vabcd B(abc) A B(adc) A (B(ad6)V B(abd)) A D(adbc) —► D(abdc). Szczerba’s [14] model of independence for Pasch’s axiom is a model of Cn(T{ US), but not of soT / ^3 - therefore the simplicity degree of £3 is 5. 6. Higher dimensions Let €'n (with n > 2) be the theory all of whose models are isomorphic to C n(F ) for some Euclidean field F. By a result of D. Scott [13], any axiom which tells us that the dimen sion is n, must have at least n + 2 variables. Therefore the simplicity degree of £'n{n > 3) is > n + 2. In fact, it is n -f 2 for all n > 3. A simple AS for £'n (with n > 2) consists of A l - A 12, A13n, A14n , A 15, where A13„. 3axa 2 :\ an+i f\ D(apaqaras), 1 A14n. V aia 2 .. .a n+2-< I /\ D(apaqara,) \ I<,<,<»+3 \ l< r < .< » + 2 69 7. Another simple axiom system Let, f.2 ho the Lpo-theory all of whoso models are isomorphic to Carte sian planes over Pythagorean ordered fields (which are ordered fields for which Vxy 3z x24-y2 =z 2 holds). The simplicity degree of £2 is 6 and a simple AS for it is the following: {(1), A 1 - A10, A12 - A H , P}, where P is Pasch’s axiom P. Vafecf/e 3 / ->((a = b)V B(abc)V B(bca)V B(cab)) B(bcd) A B(cea) B(def) A B(bfa). In order to prove that the simplicity degree of £ 2 is not less than 6, we consider the Pasch-free geometry £% = C n ((l), A1 - A 14), for which the following representation theorem holds (cf. [15]): M G M od (f- ) iff M ~ C 2(F ), where F is a Pythagorean semi-ordered field. This means that the semi order relation > satisfies the axioms (i) V i 1 > 0 V —x > 0, (ii) Vx x > 0 A —x > 0 —♦ x = 0, (iii) Vxy x > 0 A y > 0 —+x + y>0, blit need not satisfy (iv) Vxy x > 0 A y > 0 —► xy > 0, which would produce an order. We introduce a semi-norm in F by f x, if x > 0 x = < \ —x, if —x > 0 and put (for a, 6, c, d £ F 2) Bp(abc) ifT a / 6,6 ^c and ||a — 6|| 4- ||6 — c|| = ||a — c| and Df(abcd) ifT a / 6 and ||a — 6|| = ||e — rf||, where ||(*i,*2)||= \A? + A • All the 4-point sentences true in £2 are also true in £^ , therefore £-2 cannot have simplicity degree 4. A thorough analysis of all 5-point, sentences true in £2 but not in £^ (we have to consider only sentences dealing with 5 points, at. least, three of which are collinear, since all the Lp-sentences true in £2 are also true in £ 2~ (cf. [12], [17]); furthermore, either one of the other two points must be collinear with the first three or both must be collinear with one of the first three in order to obtain a 5-point sentence not. true in £ ^ shows ) that these are 70 A. Vabcde B(acb) A B(adb) A D(aead) A D(bebc) A ->(c = d) —> B(acd) which is the triangle inequality; B. Va6crfe ->(d = a) A B(bac) A D(a6ac) A D(dbdc) A D(aebd) A (B(ade) V B(aed) V (e = d)) —* B(ade) C. Vabcde ->(d = a) A fl(6ac) A D(abac) A D(dbdc) A D(aebd) A (B(abe) V B(aeb) V (e = 6)) —► B(abe) both stating that the hypotenuse is greater than the other sides of a right-angled triangle. By Satz 2.3 of [8], page 89, C n (£ ^ U {.4}) = C n (£^ u {B}) = Cn(£^~ U {C}) = C n (£ ^ U {G P}),GP where is a sentence stating that the footpoint of the altitude of a right-angled triangle lies between the endpoints of the hypotenuse (cf. [4]). The representation theorem for Cn{£^\j{GP}) is: M € Mod(£2~ U{GP}) iff M ~ C 2(F), where F is a quadratically semi-ordered Pythagorean field, i.e. the semi- order > satisfies (i), (ii), (iii) and (iv') Vxy x > 0 —► xy2 > 0. A Pythagorean field with (i), (ii), (iii), (iv') need not satisfy (iv) (cf. [4], [8]), thereforeC n(£% U {GP}) C £2, the inclusion being strict. So the only 5-point sentences true in £2 but not in £^ are A, B ,C , which are weaker than P\ therefore the simplicity degree of £2 is 6. Nevertheless, Euclidean geometry over Pythagorean ordered fields be comes a theory with simplicity degree 5, provided that we enlarge the language Lbd with the quaternary predicate symbol /. The simple AS for this theory is built up of all the axioms for £^ and Dfi. Vabcd 3e I(abcd) —► B(aeb) A B(ced), Df2. Vabcde B(aeb) A B(ced) —+ I(abcd), P'. Vabcde -•((a = c)V B(abc)V B(bca)\/ B(cab)) A B(aeb) A B(bcd) —» I(aced). This is, of course, not an AS for £2 , because £ 2 is an />BD-theory, but it is an AS for a theory which is ‘synonymous’ (cf. [1]) for the exact meaning of this term) with £2 . Thus we see that there is a theory, synonymous with £2 , with simplicity degree less than that of £2 . Hence the question: Is there any theory (expressed in a language without function symbols), synonymous with £'n(n > 2), having lower simplicity degree than that of £'n? 71 For n > 3 thr answer is negative (cf §6). For n = 2 the question remains open. Another open problem is whether the AS’s proposed for £\ and £ 2 are completely independent or not. 8. Another side of ‘simplicity’ A finitely axiomatizable theory may be axiomatized by a single sen tence (which is built up as the conjunction of all the axioms of one of its finite AS’s). Therefore, we may be interested in the minimum number of variables a single sentence axiomatizing£ 2 (or £2) has to contain. It turns out to be 6 for both £ 2 and £2. For £'2 it is Vabcde 3f (B(abc) D(cba)) A (D(abd) A B(bcd) — B(abc)) A f~*(c = d)A((B(abc)AB(abd))\/(B(abc)AB(dab))v(B{bca)AB(bda))) —*• B(acd) V B(cda) V B(dac)^j A (D(abcd) —♦ -<(a = b) A ->(c = d)) A ( D(abcd) A D(cdce) —► D(abce)) A ( D(abac) A D(acde) —► D(abde)) A (D(abcd) A D(cdae) —► D(abae)) A (B(cad) A D(abad) A B(cae) A D(abae) —► d = e) A^B(abc)A(B(ade)VB(aed))AD(abad)AD(acae) —► B(adc)AD(bcde)^j A (-i(c — d) A D(acad) A D(bcbd) A B(abc) —► D(eced)) A ^D(abad) A ( ( B(abc) A B(ade)) V (B(cab) A B(rad))) A D(acae) —* D(dcbe)j A (->(rf = e) A -i(a = b) A ->(6 = c) A ->(c = a) A D(adac) A D(bdbe) A D(cdce) —► B(abc) V B(bca) V /?(ca6)) A ^""'((« = 6)V (6 = c) V (c = a) V B(abc) V Z?(6ca) V B{cab)) -.£>(/<./») A D(/6/c)) A ((((a = 6) A -*((d = 6) —* R(dbf) A D(acdf)} A (->(« = 6) A (a = c) —+ B(afb) A D(fafb)) A ((a = 6) A (6 = c) — --(/ = a)) A ( B(acb) - -,((/ = a) V ( / = b) V B(a/6) V B(fba) V fl(6a/ ) ) ) 72 and for £ 2 it is Vabcde 3 / (B(abc) —► B(cba)) A (B(abd) A B(bcd) —*• B(abc)) a(--(c = d)A((Z?(a&c)A£(a&d))v(fl(a6c)Afl(da&))v(fl(&ca)A#(6 —► B(acd) V B(cda) V 5(dac)j A (D(abcd) —► -«(a = 6) A -»(c = rf)) AD(abcd ( ) A D(cdce) —* D(abce)) A (D(abac) A D(acde) —► D(abde)) A ( D(abcd) A £)(cdae) —*• D(a6ae)) A (#(cad) A D(abad) A Z?(cae) A D(abae) —► d = e) A ^i?(afec)A(B(a A D(acae) —► D(dcbe)j A (~<(d = e) A ->(a — b) A ->(b — c) A ->(0 = a) A D(adae) A D(bdbe) A D(cdce) —► B(abc) V B(bca) V B(cab)) A ^->((a = 6) VB(abc) V B(bca) V B(cab)) A B(bcd) A B(cea) - A S(t/a)) A ((a = 6) - -,(/ = a)) A (-^((6 = c) V (c = d) V (d = 6) V fl(tcd) V B(cdt) V B( ^ D (fb fc) A D(fcfd)) A (->(a = 6) A (6 = c) A ->( Using a theorem of Taimanov [19] (cf. also [5], Theorem 6 on page 325), we can prove that £ 2 *9 n°t axiomatizable by a single 5-variable sentence. 73 Re f e r e n c e s 1. 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Szmielew, Concerning the order and the semi-order of n-dimensional Eu clidean space, Fund. Math. 107 (1980), 47-56. 18. W . Szmielew, From Affine to Euclidean Geometry, PW N ., Warsaw, D. Reidel, Dordrecht, 1983, pp. 110-125. 19. A. D. Taimanov, Harakteristika konecno-aksiomatiziruemyh klassov modelei, Sibirsk. Mat. 2 (1961), 759-766. 20. A. Tarski, What is elementary geometry?, The Axiomatic Method, North-Holland, Amsterdam, 1959, pp. 16-29. 21. G. Weaver, Finite partitions and their generators, Zeit.schr. f. math. Logik und Gnindlagen d. Math. 2 0 (1974), 255-260. Victor Pambuccian, Current Address Str. Viitt.orului 26, Mountain House School 70266 Bucuresti 9, 12 Lake Placid Club Drive, Lake Placid NY 12946, ROMANIA. U.S.A. 74