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Computable Set Theory* The Inst of Natural Sciences Nihon Univ Proc of The Inst of Natural Sciences No 31 (1996) pp 153-157 Computable Set Theory* Masamichi WATE (Received October 31, 1995) abstract: The purpose of this paper is to make a countable model of ZFC+V=L. 1 Gddel numbering of formulas The forma,1 symbols which are used in the language ofthe set theory ZFC, are =, e, -, A. V, (, ), x, l(.c.f. [2] and [4]). A variable xn is expressed by x (n times ). The other logical connectives and quantifier are understood as the a.bbreviations, for example, the implication a -~ b is the a,bbreviation of -(a A =b) (cf. [5]). The set of all formulas used in the set theory ZFC is a cont,ext- free langua,ge. A cont,ext,-free language L(G) is generated by a cont,ext-free grammar G which is a quadruple of a finite set, V of va,riables, a finite set T of tenTtinals, a finite set P of product,ions from one varia,ble t,o a st,ring in (V U T)*, and a start symbol S (cf. [3]). Let G (VT P ~) where V {S X} T {x, ,=,e,-,A,V,(,)} and P = {S -~ ~(S), S -~ (S) A (S). S - VX(S), S --~ X = X, S -=} X e X, X -~ X , X .-~ x). Then L(G) is the set of all formulas used in ZFC. For example, the axiom of empty set lxo(Vxl(-(xl e xo))) is derived from S in G as follows: S => -(S) => -(VX(S)) => -(Vx(S)) :;> ~(Vx(-(S))) i.e. 3x(S) => ~x(VX(S)) ::> 3x(VXl(S)) => 3x(Vx (S)) ~> ~x(Vxl(=(S))) => ~x(Vx (=(X ~ X))) => Sx(Vx (-(X e X))) => Sx(Vx (-(x e X))) => Sx(Vx (=(x e x))) i.e. 3xo(Vxl(-(xl e xo))) Now, we carry out, the G6del numbering of the formulas in ZFC according to the order of the derivation in t.he context-free_ grammar G. Whiie a G6del numbering of a certain set is an onto mapping from a certain subset of the natural numbers to the set, i.p_. e.ach member of the set corresponds to at least one natural number, and ea,ch natural number corresponds to at most one member of the set (cf.[5]). Whe_n a, nat,ural number n corresponds t,o a member a of the set, we call that 71, is a, G6del number of a., and we write n = gn(a) or a := ng(n). So, we define gn(:=) = 3, gn(e) =: 5, gn(-) ~ 7, gn(A) = 9, gn(V) = 11, gn(x) =: 13, gn( ) = 15. The G6del number of a variable xn is gn(xn) = P gn(x)pgn(j)o I ' gn(1)' '_ 213315Pn . -. .p~5' where pn is the (n + 1)-t,h prime nmTiber. The Gbdel numbers of formulas is defined induct,ively, that is gn(xn e xm ) = 2gn(e)3gn((c')5gn(ir) = 253gn(x.)5gn(x~). Similarly, gn(x = xm) = 233gn(e~)5gn('r~) n , *1991 M(~'thematics S'!'bject Ciassification' Primary 04-04, c4A25. Key t('ords and phrases. ZFC, partia' recursive. G5del number' 153 - ( 13 ) Masamichi WATE gn(-(A)) = 273g"(A), gn((A) A (B)) = 293g"(A)5g"(B), gn(Vx~(A)) = 2113~"('.)5g"(A). 'Then, a Gbdel number of each formula is determined uniquely. 2 G6del numbering Of the partial recursive functionS We define (n'o, ' ' ' , nk) = 2"" . .P~k (cf. [6] and [7]). By this notation, the G6del numbering in the previous section turns out gn(x~) ;~ (13, 15, . , 15), gn(xn e xm) = (5,gn(xn)'gn(xm))' and so on. A function f from the n-tuples of the natural numbers to the natural numbers is partial recursive if it is got,t,on by finite applications of the following (1) - (6) (cf. [5]). (1) C(x)c~ O, (2) S(x) c~: x + 1, (3) Ut~(xl, . ,x..) c:~ xi (1 < i < n), (4) f(xl, . , x,*) c~ h(91(xl, . ,x~), . ,9*(xl, . ,. x~. )) when 91 ' ' ' " 9m and h are partial recursive, f(O x2, ' ' ' ,x..) c:~ 9(x2, ' ' "x~) (5) f '(xl + 1, x2, ' ' ' , x~) ~~ h(f (xl, x2' ' ' " x~), xl , x'_,.. , x~) when g and h a,re partial recursive, (6) f(xl , . , x~) c:( /1y (g(y,xl, . , x~) = O) when 9 is partial recursive. c:~ mea'ns t,ha,t, both sides have the same domain and have the same va,lue for an a'rgument in t,he domain. Moreover, ,1y (g(y,xl, . ,x~) = O) means rrrinimum natural number y such that g(y, xl , . , xn) = O for :~'l , . ,xn' (cf. [5]) Now, we c.arry out, the Gbdel numbe.ring of the partial recursive functions according to the order of the construction of them. We define (i) gn(C) = (1), (ii) gn(S) = (2), (iii) gn(UsP) = (3, n, i), (iv) gn(f) = (4, gn(91), . , gn(9~), gn(h)) when f is of the form (4), (v) gn(f) = (5,gn(9), gn(h)) when f is of the form (5), (vi) gn(f) = (6,gn(g)) when f is of the form (6). For examp]e, in t,he case of f(x) c:~ 1, gn(f) = (4, (1), (2)), since f(x) ~: S(C(x)), and in the case of f(x,y) ~~ x + y, gn(f) = (5, (3, 1, 1), (4, (3, 3, 1), (2))), since f is gotten by the applicat,ion of the form (5) for 9(y) c~: Ui(y) and h(z,x,y) ~: S(Uf(z,x,y)). Bot,h of (1) and (4, (3, 1, 1), (1)) are G6del nulnbers of the constant function f(x) c:~ O, since C(Uf(x)) c:~ C(x) ~~ O. Thus, a C'.6del number of a partia,1 recursive function is determined but not uniquely. This technic of G6del numbering owes [6] and [7]. ( 14 ) - 154 Computable Set Theory 3 G6del numbering of sets and axioms of ZFC Definition Let, f be a, pa.rtial recursive function with one va,riable and the domain of it be an initial segment of c,). If the range of f is a family of Gddel numbers of the me.mbers of a set a, a C...'6del nuniber of f is a G6del number of a. Namely, when f(x) c~ {e}(x), gn(a) = e or ng(e) = a. The notat,ion {e} owes [5], and it means that e is a G6del number of f. In this way, G6del numbers of sets are defined inductively, but it is impossible to do G6del nmTrbering about all sets. Now, we consider the family ZFCre of all sets which have G6del numbers. It is familiar by the L6wenheim-Scolem theorem [8] and [9] that there is a. countable model of ZFC. 110wever, we can see that ZFCre is not only a countable model, but also an effective model of ZFC. The-orem I ZFCre sat,isfies all axioms of ZFC except the power set and the replacement. Proof Axiom of Empty (6, (4, (3,2, 1), (2))) is a G6del number ofc, since f(x) c:~ uy (S(U~ (x, y)) :: O) is undefined for all x. Axiom of Pairing For any two sets a and b with G6del numbers, put f(x)- gn(a) ifx=0 gn(b) if x ;= 1 Then f is pa,rtial recursive, and gn(f) := gn({a, b}). Axiom of Extensionality a = b and one of them has a G6del number if and only if a, and b have t,he same_ G6del numbers. Axiom of Union Let e be a G6del number of a. At stage 1, we compute {{e}(O)}(O) one step. At st,age 2, we compute {{e}(O)}, {{e}(O)}(1),{{e}(1)}(O) two steps. Generally at stage n, we cornpute {{e}(O)},{{e}(O)}(1), {{e}(1)}(O), ... ,{{e}(O)}(n),. .. ,{{e}(.n)}(O) n steps. We assign the values of ones of which computation are accomplished in turn, to f(O), f(1), . .. Then the function f is partia,1 recursive by Church's Thesis [1] and the domain of it is an initial segment of~). And gn(f) is a G6del number of Ua. Axiom of Infinity Put f(O) :: gn(c) and f(n+ 1) =: gn(ng(f(n)) U {ng(f(n))}). Then, f is re-cursive and f(n) = gn(n) for all n e e'). (n in f(n) means a natural number in the usual sense, and n, in gn(n.) means a natura,1 number in the set theoritical sense, i.e. O ::: c and n+ I = nU{n}.) Axiom of Regularity It is shown e.asily by the induction on e that ZFCre is embedded into V~I. Ile.nce, ZFCre is well-ordered set with respect to e, since V~'l is s0. Therefore, any non-empty set a of ZFC,.e has a minimal element x with respect to ~E in it. Clearly, a n x = c. Axiom of Choice Let a be a non-empty set in ZFCre of which each element is non-empt,y. I.et, e be a G6del number of a. A G6del number of an element of a is of t,he form {e)(n). Hence, {{e}(n)}(O) is a G6del number of one element of ng({e}(n)). C] Let A(x, y, xl, . , xtl) be a fonnula such that Vx]!yA(x, y, cl, . , cn') for constants cl ' ' ' . I cn ' Put,t,ing y = f(x) if and only if A(x, y, cl' ' ' " cn)' f is a function from the class of all sets V to V.
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