The Axiom of Choice and the Law of Excluded Middle in Weak Set Theories

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1 The Axiom of Choice and the Law of Excluded Middle in Weak Set Theorie John L. Bell Department of Philoophy, Univerity of Wetern Ontario In contructive mathematic the axiom of choice (AC) ha a omewhat ambiguou tatu. On the one hand, in intuitionitic et theory, or the local et theory aociated with a topo ([2]) it can be hown to entail the law of excluded middle (LEM) ([ 3 ], [ 5 ]). On the other hand, under the propoition-a type interpretation which lie at the heart of contructive predicative type theorie uch a that of Martin-Löf [9], the axiom of choice i actually derivable (ee, e.g. [11] ), and o certainly cannot entail the law of excluded middle. Thi incongruity ha been the ubject of a number of recent invetigation, for example [6], [7], [9], [12]. What ha emerged i that for the derivation of LEM from AC to go through it i ufficient that et (in particular power et), or function, have a degree of extenionality which i, o to peak, built into the uual et theorie but i incompatible with contructive type theorie Another condition, independent of extenionality, enuring that the derivation goe through i that any equivalence relation determine a quotient et. LEM can alo be hown to follow from a uitably extenionalized verion of AC. The argument etablihing thee intriguing reult have motly been formulated within a type-theoretic framework. It i my purpoe here to formulate and derive analogou reult within a comparatively traightforward et-theoretic framework. The core principle of thi framework form a theory weak et theory WST which lack the axiom of extenionality 1 and upport only minimal et-theoretic contruction. WST may be conidered a fragment both of (intuitionitic) 0 -Zermelo et theory and Aczel contructive et theory ([1]). In particular WST i, like contructive type theorie, too weak to allow the derivation of LEM from AC. But we hall ee that, a with contructive type theorie, beefing up WST with extenionality principle (even very moderate one) or quotient et enable the derivation to go through. Let L be the firt-order language of (intuitionitic) et theory which, in addition to the uual identity and memberhip ymbol = and alo contain a binary operation ymbol 1 Set theorie (with claical logic) lacking the axiom of extenionality eem firt to have been extenively tudied in [4] and [10].

2 2, permitting the formation of ordered pair 2. At certain point variou additional predicate and operation ymbol will be introduced into L. The retricted quantifier x a and x a are defined a uual, that i, a xx ( a...) and xx ( a...) repectively. A formula i retricted if it contain only retricted quantifier. Weak et theory WST i the theory in L with the following baic axiom (in which the free variable are undertood to be univerally quantified, and imilarly below): Unordered Pair u x[ x u x = a x = b] Ordered Pair ab, = cd, a= c b= d Binary Union u x[ x u x a x b] Carteian Product u x[ x u y a z b ( x = y, z )] Retricted Subet u x[ x u x a ϕ ] where in thi lat axiom ϕ i any retricted formula in which the variable u i not free. We introduce into L new predicate and operation ymbol a indicated below and adjoin to WST by the following definitional axiom: a b x[ x a x b] a b x[ x a x b] Ext( a) x a y a[ x y x = y] x a b x a x b x { a, b} x = a x = b { a} = { a, a} x r y x, y r y { x a : ϕ( x)} y a ϕ( y) ( ϕ retricted) x 0 1 = {0} 2 = {0,1} x a b u a v b( x = u, v ) x a + b u a v b[ x = u,0 x = v,1 ] f: a b f a b x a y bxf ( y) x y z[( xf y xf z) y= z] Fun( f ) a b( f : a b) f : a b x a x f f( x) f : a b g : b c g f : a c x a[( g f )( x) = g( f( x))] f : a b f : a b y b x a[ y = f( x)] π : a + b a b x a[ π ( x,0 ) = x] y b[ π ( y,1 ) = y] π : a + b 2 x a[ π ( x,0 ) = 0] y b[ π ( y,1 ) = 1] Eqa (, ) a a x axx ( ) x a y axy ( yx) x a y a z a[( xy yz) xz] Compr (, ) x x y[( xx x ry) xry] Comp() r x x y[( x x x r y) x r y] Extn( f, ) Fun( f ) x x [ x x y y ( x f y x f y ) f ( x) = f ( x )] Ex( f ) Fun( f ) x x [ x x y y ( x f y x f y ) f ( x) = f ( x )] 2 While the ordered pair <u,v> could be defined in the cutomary way a {{u}, {u,v}}, here it i taken a a primitive operation a it i in type theory both for reaon of implicity and to emphaize the fact that for our purpoe it doe not matter how (or indeed whether) it i defined et-theoretically.

3 3 Mot of thee definition are tandard. The function π and 1 pair onto their 1 t and 2 nd coordinate repectively: clearly, for u, v a + b we have (proj) u = v [ π 1( u) =π1( v) π 2( u) =π 2( v)]. π are projection of ordered 2 The relation i that of extenional equality. Ext(a) expree the extenionality of the member of the et a. Eq(,a) aert that i an equivalence relation on a. If r i a relation between a and b, and an relation on a, Comp(r,) expree the compatibility of r with, and Comp(r) the compatibility of r with extenional equality. If f: a b, and i an equivalence relation on a, Etxn(f,) expree the idea that f treat the relation a if it were the identity relation: we hall then ay that f i - extenional. Ex(f) aert that f i extenional in the ene of treating extenional equality a if it were identity. In addition to the axiom of WST, We formulate the following axiom additional to thoe of WST (recalling that in tating axiom that all free variable are univerally quantified): : Extenionality Extub(1) Extub(2) Ext({a, b}) a 1 b 1 Ext({a, b}) a 2 b 2 Ext({a, b}) While Extenionality aert that all extenionally equal et are identical, Extub(1) and Extub(2) are weak verion confining the aertion jut to ubet of 1 or 2. Explicitly, Extub(1) and Extub(2) aert that each doubleton compoed of ubet of 1, or of 2, i extenional. Notice that ince 1 2, Extub(1) i a conequence of Extub(2). Following [1], we define a et a to be a bae if every relation with domain a include a function with domain a, i.e. Bae(a) b r[ r a b x a y b( x r y) f : a b x a( x r f( x))]. We call a an extenional bae if every relation with domain a compatible with extenional equality include an extenional function with domain a, i.e. Extbae(a): b r[ r a b Compr ( ) x a y b( x r y) f : a b[ Ex( f ) x a( x r f( x))]]

4 4 We ue thee notion to tate a number of verion of the axiom of choice (again recalling that in tating axiom that all free variable are univerally quantified): (Intenional) Axiom of Choice AC Bae(a) Weak Axiom of Choice 1 WAC(1) a 1 a 1 b 1 b 1 Bae( { aa,, bb, }) Weak Axiom of Choice 2 WAC(2) a 2 a 2 Bae( { aa, }) Univeral Extenional Axiom of Choice UEAC Eq(, a) r a b Comp( r, ) x a y b( x r y) f : a b[ Extn( f, ) x a( x r f ( x))] Extenional Axiom of Choice EAC Extbae(a) In aerting that every et i a bae, AC mean, a uual, that a choice function alway exit under the appropriate condition on an arbitrarily given relation. WAC(1) and WAC(2) retrict the exitence of uch choice function to relation whoe domain are doubleton of a certain form 3. UEAC aert that, in the preence of an equivalence relation with which a given relation r i compatible, the choice function can be taken to be -extenional. EAC i the pecial cae of UEAC in which the equivalence relation i that of extenional equality. In view of the fact that AC can be een to be the pecial cae of UEAC in which the equivalence relation i the identity relation, AC i ometime known a the intenional axiom of choice. Our next axiom i Quotient Eq(, a) u f[ f : a u x a y a[ f ( x) = f ( y) x y]] Thi axiom aert that each equivalence relation determine a quotient et. In WST + Quotient, we introduce operation ymbol i, [ i] i and adjoin the definitional axiom (Q) Eq(, a) [ x a([ x] a / r ) u a x a( u = [ x] ). x a y a[[ x] = [ y] x y]] 3 Uing the fact that ubet of 2 are in natural bijective correpondence with pair of ubet of 1, it can be hown that WAC(1) and WAC(2) are in fact equivalent. Neverthele for our purpoe it will be ueful to have both principle.

5 5 Here a i the quotient of a by and, for x a, [x] i the image of x in a/. Reminding the reader that our background logic i intuitionitic, we finally introduce the following logical cheme: Retricted Excluded Middle REM ϕ ϕ for any retricted formula ϕ We are going to prove the following reult. Theorem 1. REM i derivable in: (a) WST + Extub(1) + WAC(1); (b) WST + Extub(2) + WAC(2); and (c) WST + EAC. Theorem 2. REM i derivable in WST + AC + Quotient Theorem 3. AC UEAC i derivable in WST + Quotient. Thu, while in the abence of extenionality for doubleton of ubet of 1 or 2, the intenional axiom of choice doe not entail the law of excluded middle, with that degree of extenionality the law of excluded middle become a conequence of very weak verion of the intenional axiom of choice. (A fortiori REM i derivable in WST + Extenionality + AC.) Moreover, the extenional axiom of choice entail the law of excluded middle without additional extenionality aumption. And finally, when quotient are preent the intenional axiom of choice i no weaker than it univeral extenional verion and entail the law of excluded middle without additional extenionality aumption. Proof of Theorem 1. (a) We argue in WST + Extub(1) + WAC(1). Given an arbitrary retricted formula ϕ, we define = {x {0}: ϕ} and a = {,{0}, {0}, }. It i then eaily hown that uv, a x 2[( x= 0 0 u) ( x= 1 0 v)]. So WAC(1) give f: a 2 uch that, for uv, a

6 6 (1) f ( uv, ) = 0 0 u (2) f ( uv, ) = 1 0 v. Since f map to 2, we have [ f(,{0} ) = 0 f(,{0} ) = 1] [ f( {0}, ) = 0 f( {0}, ) = 1]. From thi, together with (1) and (2), it follow that whence, uing the ditributive law, [0 f(,{0} ) = 1] [ f( {0}, ) = 0 0 ], 0 [ f(,{0} ) = 0 f( {0}, ) = 1]. Writing ψ() for the econd dijunct in thi lat formula, it become (3) 0 ψ(). Now from 1 we deduce whence uing Extub(1), 0 {0}, 0 = {0}. Hence [0 ψ( )] [ = {0} ψ( )] ψ({0}) 0 = 1. Since clearly 0 1, we conclude that ψ( ) 0 and (3) then yield (4) 0 0. But obviouly 0 ϕ, o (4) give ϕ ϕ, a required. (b) 4 We argue in WST + Extub(2) + WAC(2). Given a formula ϕ, define Since 0 a and 1 b, we have a = { x 2 : x = 0 ϕ }, b = { x 2 : x = 1 ϕ }. 4 Uing the obervation in the previou footnote that WAC(1) and WAC(2) are equivalent, it can be een that (b) actually follow from (a). However the direct proof given here, baed on that given in [5], i, in the author view, coniderably more illuminating.

7 7 x { a, b} y 2. y x, and o WAC(2) applied to the relation r = { xy, { ab, } 2: y x} ) yield a function f: {a, b} 2 for which x { a, b}. f( x) x. It follow that f ( a) a f( b) b, whence [ fa ( ) = 0 ϕ] [ fb ( ) = 1 ϕ ]. Applying the ditributive law, we then get whence ϕ [ fa ( ) = 0 fb ( ) = 1]. (1) ϕ f ( a) f( b). Now clearly ϕ a b, and from thi and Extub(2) we deduce ϕ a = b, whence (2) ϕ f ( a) = f( b). It follow that fa ( ) fb ( ) ϕ, and we conclude from (1) that ϕ ϕ, a required. (c) Here the argument in WST+ EAC i the ame a that given in (b) except that in deriving (2) above we invoke EAC in place of Extub(2). To jutify thi tep it uffice to how that Comp(r), where r i the relation defined in the proof of (b). Thi, however, i clear. Proof of Theorem 2. For b a, we ay that b i detachable (in a) if x a[ x b x b]. An indicator for b (in a) i a function g: a 2 2 atifying x a[ x b g( x,0 ) = g( x,1 )]. It i eay to how that a ubet i detachable if and only if it ha an indicator. For if b a i detachable, then g: a 2 2 defined by

8 8 g( x,0 ) = g( x,1 ) = 0 if x b g( x,0 ) = 0 g( x,1 ) = 1 if x b i an indicator for b. Converely, for any function g: a 2 2, we have g( x,0 ) = g( x,1 ) g( x,0 ) g( x,1 ), o if g i an indicator for b, we infer x a[ x b x b], and u i detachable. Now we how in WST + AC + Quotient that every ubet of a et ha an indicator, and i hence detachable. Given b a, let be the binary relation on a + a given by: = { x,0),( x,0 : x a} { x,1), x,1 : x a} { x,0), x,1 : x b} { x,1), x,0 : x b}. It i eaily checked that Eq(, a + a). Alo, it i clear that, for z, z a + a, (1) z z π 1( z) =π 1( z ) and, for x a, (2) x b x,0 x,1. Invoking axiom (Q) above, we introduce the quotient ( a + a) of a + a by and the image [u] of an element u of a + a in ( a + a) for which we then have (3) z ( a + a) u a + a( z = [ u] ) and (4) u a + a v a + a[[ u] = [ v] u v]. Applying AC to (3) yield a function (5) Clearly f i one-one, that i, we have f : ( a + a ) a + a for which z ( a + a) z = [ f( z)] ). (6) f(z) = f(z ) z = z. Next, oberve that, for i = 0, 1, and x a, (7) π 1 ([ f ( xi, )] ) = x. For from (5) we have [ xi, ] = [ f([ xi, ] )], whence by (4) xi, f([ xi, ] ). Hence by (1) 1( xi 1 f xi π, ) = π ( ([, ] )). (7) now follow from thi and the fact that π 1 ( xi, ) = x. We have alo (8) x b f([ x,0 ] ) = f([ x,1 ] ). For we have

9 9 Now define g: a 2 2 by x b x,0 x,1 uing (2) [ x,0 ] = [ x,1 ] uing (4) f([ x,0 ] ) = f([ x,1 ] ) uing (6). g, ) = π ( ([, ] )). ( xi 2 f xi We claim that g i an indicator for b. Thi can be een from the following equivalence: x b f([ x,0 ] ) = f([ x,1 ] ) (by (8)) π 1( f([ x,0 ] )) = π 1( f([ x,1 ] )) (by (proj)) π 2( f([ x,0 ] )) = π 2( f([ x,1 ] )) π ( f([ x,0 ] )) = π ( f([ x,1 ] )) (uing (7) 2 2 g( x,0 ) = g( x,1 ). So we have hown that WST + AC + Quotient every ubet of a et ha an indicator, and i accordingly detachable. Thi latter fact eaily yield REM. For, given retricted ϕ, for any a, the et b = {x a: ϕ} i then a detachable ubet of a, from which x a( ϕ ϕ) immediately follow. By taking a = {x} we get ϕ ϕ. Proof of Theorem 3. It uffice to derive UEAC from AC in WST + Quotient. Auming Eq(, a), we ue AC a in the proof of Theorem 2 to obtain a function u = [ p( u)] for all u a. From thi we deduce [ x] [ ([ ] )] = p x, whence (1) xp([ x ] ) for all x a. Auming the antecedent of UEAC, viz., Eq(, a) r a b Comp( r, ) x a y b( x r y), p : a a uch that define the relation r a b by u r y p(u) r y. Now ue AC to obtain a function g : a b for which u a ( u r g( u)), i.e. (2) u a ( p( u) r g( u)). Define f: a b by f(x) = g([x] ).

10 10 Then by (2) x a( p([ x] ) r g([ x] )). From thi, (1) and Comp(r, ) it follow that x a( x r g([ x] )), i.e. (3) x a( x r f( x)). Moreover, for all x, x a, we have x x [x] = [x ] f(x) = g([x] ) = g([x ] ) = f(x ), whence Extn(f, ). Thi, together with (3), etablihe the conequent of UEAC. Concluding remark. 1. The proof of Theorem 1 (a) i an adaptation of the proof of the analogou reult given in [2], pp , and that of (b) i baed on that given in [5]. 2. The proof of Theorem 2 i an adaptation to a et-theoretical context of the argument in [3] that, in a topo atifying the axiom of choice, all ubobject are complemented. By weakening Quotient to the aertion Quotient(1 + 1) that quotient et are determined jut by equivalence relation on the et 1 + 1, the proof of Theorem 2 how that REM i derivable in the theory WST + AC + Quotient(1 + 1). 3. Quotient can be derived within WST augmented by the full extenional power et axiom Extpow uextu [ ( ) xx [ u x a]] So (cf. [7 ]) adding extenional power et to WST + AC yield REM. 4. Another verion of the axiom of choice, eaily derivable in WST from AC i: AC* x a[ x b y( y x)] f : a b x a( f( x) x)]. If one add to WST the nonextenional power et axiom, viz. Pow u x[ x u x a], then AC* become derivable from AC. Note that while Extpow entail REM, Pow i logically harmle, that i, it ha no noncontructive logical conequence uch a LEM. The extenional verion of AC*, viz. EAC* x a[ x b y( y x)] f : a b[ Ex( f ) x a( f( x) x)]. i derivable in WST from EAC. In WST + Pow, EAC and EAC* are equivalent.

11 11 5. A verion of the axiom of choice conidered in [9] i what we hall call Rep(reentative) Eq(, a) f[ f : a a x a( xf ( x)) x a y a[ xy f ( x) = f ( y)]]. Rep aert that unique repreentative can be choen from the equivalence clae of any equivalence relation. Obviouly, in WST, Rep implie Quotient. Moreover, the proof of Theorem 2 i eaily adapted to how that, in WST, Rep yield REM. In WST, AC + Quotient entail Rep, and, in WST + Pow, converely. 6. Finally, conider the following verion of AC which are cloely related to that introduced by Zermelo [13] (ee alo [9]), namely ACZ [ x a y( y x) x a y a[ z( z x z y) x = y]] u x a! y( y x y u ) EACZ [ x a y( y x) x a y a[ z( z x z y) x y]] u x a! y( y x y u ) Thee are contrual of the aertion that, given any collection of mutually dijoint nonempty et, there i a et interecting each member of the collection in exactly one element. Clearly EACZ implie ACZ; the former i readily derivable from EAC and the latter from AC. Since REMS i not a conequence of AC, it cannot, a fortiori, be a conequence of ACZ. But, like EAC, EACZ can be hown to yield REM. We ketch the argument, which i imilar to the proof of Thm. 2(b). Given a retricted formula ϕ, define b = { x 2: x = 0 ϕ }, c = { x 2: x = 1 ϕ } and a = {b, c}. A traightforward argument how that a atifie the antecedent of EACZ. So, if thi lat i aumed, it conequent yield a u with exactly one element in common with b and with c. Writing d and e for thee element, one eaily how that (*) ϕ d e. Now ince it i alo eaily hown that ϕ d = e, it follow that d e ϕ, and thi, together with (*) yield ϕ ϕ.

12 12 Acknowledgement I am grateful to the (anonymou) referee for making a number of helpful uggetion for harpening the paper. Reference [1] Aczel, P. and Rathjen, M. Note on Contructive Set Theory. Intitut Mittag-Leffler Report no. 40, 2000/2001. Available on firt author webite. [2] Bell, J. L. Topoe and Local Set Theorie: An Introduction. Clarendon Pre, Oxford, To be republihed by Dover, [3] Diaconecu, R. Axiom of choice and complementation. Proc. American Mathematical Society 51, 1975, pp [4] Gandy, R.O, On the axiom of extenionality, Part I, Journal of Symbolic Logic 21, 1956, pp ; Part II, ibid., 24, 1959, pp [5] Goodman, N. and Myhill, J. Choice implie excluded middle. Z. Math Logik Grundlag. Math. 24, no. 5, 1978, p [6] Maietti, M.E. About effective quotient in contructive type theory. In Type for Proof and Program, International Workhop Type 98, Altenkirch, T., et al., ed., Lecture Note in Computer Science 1657, Springer-Verlag, 1999, pp [7] Maietti, M.E. and Valentini, S. Can you add power-et to Martin-Löf intuitionitic type theory? Mathematical Logic Quarterly 45, 1999, pp [8] Martin-Löf, P. Intuitionitic Type Theory. Bibliopoli, Naple, [9] Martin-Löf, P. 100 year of Zermelo axiom of choice: what wa the problem with it? The Computer Journal 49 (3), 2006, pp [10] Scott, D. S. More on the axiom of extenionality. In Eay on the Foundation of Mathematic, Magne Pre, Jerualem, 1966, pp [11] Tait, W. W. The law of excluded middle and the axiom of choice. In Mathematic and Mind, A. George (ed.), pp New York: Oxford Univerity Pre, [12] Valentini, S. Extenionality veru contructivity. Mathematical logic Quarterly 42 (2), 2002, pp [13] Zermelo, E. Neuer Bewei für die Möglichkeit einer Wohlordnung, Mathematiche Annalen 65, 1908, pp Tranlated in van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic , Harvard Univerity Pre, 1967, pp