INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.

Transcription

1 INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton of reducblty between sets of natural numbers based on ther computatonal strength. In the past years many reducbltes, together wth ther nduced degree structures have been nvestgated. One of the aspects n these nvestgatons s always the characterzaton of the theory of the studed structure. One dstngushes between global structures, ones contanng all possble degrees, and local structures, contanng all degrees bounded by a fxed element, usually the degree whch contans the haltng set. It has become apparent that modfyng the underlyng reducblty does not nfluence the strength of the frst order theory of ether the nduced global or the nduced local structure. The global frst order theores of the Turng degrees [15], of the many-one degrees [11], of the 1-degrees [11] are computably somorphc to second order arthmetc. The local frst order theores of the computably enumerable degrees 1, of the many-one degrees [12], of the 0 2 Turng degrees [14] are computably somorphc to the theory of frst order arthmetc. In ths artcle we consder enumeraton reducblty, and the nduced structure of the enumeraton degrees. Enumeraton reducblty ntroduced by Fredberg and Rogers [4] arses as a way to compare the computatonal strength of the postve nformaton contaned n sets of naturals numbers. A set A s enumeraton reducble to a set B f gven any enumeraton of the set B, one can effectvely compute an enumeraton of the set A. The nduced structure of the enumeraton degrees D e s an upper semlattce wth least element and jump operaton. Ths structure rases partcular nterest as t can be vewed as an extenson of the structure of the Turng degrees. There s an somorphc copy of the the Turng degrees n D e. The elements of ths copy are called the total enumeraton degrees. The enumeraton jump operaton gves rse to a local substructure, G e,consstng of all degrees n the nterval enclosed by the least degree and ts frst jump. Cooper [1] shows that the elements of G e are precsely the enumeraton degrees whch contan Σ 0 2 sets, or equvalently are made up entrely of Σ 0 2 sets, whch we call Σ 0 2 degrees. Ths structure can n turn be vewed as an extenson of the structure of the 0 2 Turng degrees, whch s somorphc to the Σ 0 2 total degrees. Slaman and Woodn [16] prove that the theory of the global structure of the enumeraton degrees, D e, s computably somorphc to the theory of second order arthmetc and show that the local theory s undecdable. In the same artcle The authors were partally supported by BNSF through contract D / The second author has been supported by a Mare Cure European Rentegraton Grant No wthn the 7th European Communty Framework Programme. 1 Ths result s due to Harrngton and Slaman, see [13] for a publshed proof. 1

2 2 H. GANCHEV AND M. SOSKOVA under queston 3.5 they ask whether or not the theory of the local structure, G e, s computably somorphc to the theory of frst order arthmetc. The same queston appears frst n [2] n 1990, and then agan n other artcles, see [17], [3] and [18]. Recently Kent [10] has characterzed the frst order theory of a substructure of G e. He proves that the frst order theory of the 0 2 enumeraton degrees s computably somorphc to the theory of true arthmetc. However as there are properly Σ 0 2 enumeraton degrees, degrees whch contan no 0 2 set, ths does not settle the problem for the theory of G e. The goal of ths artcle s to gve an answer to ths longstandng queston. Note that snce the degrees n G e are the degrees of the Σ 0 2 sets we can assocate a natural number a to every a G e n such a way that the relaton on natural numbers, defned by a b a b, s arthmetc. Thus there s a computable translaton φ of every sentence of G e nto a sentence of arthmetc such that G e = θ N = φ(θ). For the converse drecton t s enough to show the exstence of a class of defnable wth parameters structures N (q) n G e and a formula SMA, such that the followng two statements are true. (1.1) If G e = SMA(q), then N (q) s a standard model of arthmetc. (1.2) There are parameters q, such that G e = SMA(q). The methods used to prove ths result rely on the noton of a K-par, ntroduced by Kalmulln [9] and used to show the defnablty of the enumeraton jump operaton. In [5] we show that K-pars are frst order defnable n G e. In Secton 3 we use ths result to mprove the local verson of Slaman and Woodn s codng lemma [16], showng that a larger class of relatons can be coded wth parameters n G e. Usng ths result we prove the exstence of a formula SMA satsfyng (1.1) and (1.2). Fnally we show the exstence of a parameterless nterpretaton of true arthmetc. 2. Prelmnares In ths secton we shall ntroduce all the notons and results that wll be needed throughout the paper. We start wth the noton of enumeraton reducblty. As noted above, ntutvely a set A s enumeraton reducble to a set B, denoted by A e B f and only f there s an algorthm transformng every enumeraton of B nto an enumeraton of A. Formally A e B [A = W (B)], where W s the c.e. set wth Gödel ndex and W (B) stands for the set W (B) = {x x, u W & D u B}, where D u s the fnte set wth canoncal ndex u. The relaton e s reflexve and transtve (but not antsymmetrc) and so t gves rse to a nontrval equvalence relaton e defned by A e B A e B & B e A. We denote d e (A) = {B A e B}. The equvalence classes under e are called enumeraton degrees. We shall denote by D e the collecton of all enumeraton degrees. The preorder e on sets nduces a partal order on degrees, defned by a b A a B b[a e B].

3 INTERPRETING TRUE ARITHMETIC IN G e 3 The degree 0 e consstng of all c.e. sets s the least degree n D e. Furthermore the degree d e (A B), where A B = {2x x A} {2x + 1 x B}, s the least degree whch s greater then or equal to the degrees d e (A) and d e (B). Thus D e = (D e, ) s an upper semlattce wth least element. Besdes the jon of two set, we shall need the noton of a unform jon of a system of sets. Let I N and let {X I} be a system of sets of natural numbers. We defne the unform jon of the system to be X = { x, x X, I}. I The unform jon of a system {X I} s the least unform upper bound for t, n the sense that X e A unformly n and I e A f and only f I X e A. Note that for a fnte system {X n} we have n X e X 0 X 1 X n. We ntroduce the enumeraton jump on sets by settng A = K A (N \ K A ), where K A = <ω W (A). The jump operaton on sets has the property A e B A e B and hence we can defne a jump operaton on degrees by settng d e (A) = d e (A ). Furthermore A e A and A e A, so that the jump operaton on degrees s strctly monotone. Ths paper s dedcated to the degrees that le between the least degree 0 e and ts frst jump 0 e,.e. the degrees n the nterval [0 e, 0 e]. We shall denote by G e the substructure G e = ([0 e, 0 e], ) of D e. The theory of G e s referred to as the local theory of the enumeraton degrees. As noted above, Cooper [1] has shown that the degrees n G e are exactly the degrees of the Σ 0 2 sets, so that the local theory of the enumeraton degrees s actually the theory of the Σ 0 2 enumeraton degrees. A specal role n ths paper shall be played by the so called low degrees. A degree a s low f and only f a = 0 e. The sets contaned n a low degree are called low sets. Thus A s a low set f and only f A e. The low sets have the followng characterzaton: A s low W (A) s a 0 2 set for every. In partcular every low set s a 0 2 set. An nstance of low degrees (and sets) are the so called Kalmulln pars, or brefly K-pars, of degrees n G e (or Σ 0 2 sets). We say that the par of sets {A, B} s a K-par f both A and B are not c.e. 2 and A B W, A B W for some, where A B = { a, b a A, b B}. We say that the par of degrees {a, b} s a K-par, f there are A a and B b, such that {A, B} s a K-par of sets. The followng propertes proved by Kalmulln [9] shall be mportant for us: (1) Let {a, b} be a K-par and a, b 0 e. Then both a and b are low. (2) Let {a, b} be a K-par. Then for every x x = (a x) (b x). In partcular every K par s a mnmal par. Furthermore f a 1 a, b 1 b and {a, b} s a K-par, so s the par {a 1, b 1 }. 2 In the orgnal defnton ths s not requred, but t s useful for our goals.

4 4 H. GANCHEV AND M. SOSKOVA In [5] we have seen that there s a formula LK(a, b) that locally defnes the K-pars,.e. for arbtrary a, b G e {a, b} s a K-par G e = LK(a, b). Thus we have a frst order defnable class of low degrees n G e, namely the class of the degrees n G e that are part of K-pars. Snce Jockusch [7] has proved that each total degree s the least upper bound of a K-par, ths class s not empty. Low degrees allow us to encode a specal knd of antchans,.e. sets of parwse ncomparable degrees. Consder the formula SW (z, a, p 1, p 2 ) defned by z a & z (z p 1 ) (z p 2 ) & y < z [y = (y p 1 ) (y p 2 )],.e. z s less or equal to a and t s a mnmal soluton to x (x p 1 ) (x p 2 ). Obvously the set Z(a, p 1, p 2 ) = {z G e G e = SW (z, a, p 1, p 2 )} s an antchan for every choce of a, p 1 and p 2 n G e. On the other hand t s not known whether every antchan n G e can be encoded by the formula SW. However, the followng result of Slaman and Woodn [16] wll be enough for our purposes: Theorem 1 (Slaman, Woodn [16]). Let k ω. Let A be a low set and let {Z < k} be a system of ncomparable reals (.e. for each j, Z e Z j and Z j e Z ) unformly e-reducble to A. Then there are degrees p 1, p 2 G e, such that for all z G e = SW (z, d e (A), p 1, p 2 ) z = d e (Z ) for some < ω. The noton of a K-system ntroduced n [6] s a natural extenson of the noton of a K-par. We say that the fnte system of sets {A 1,..., A n } (degrees {a 1,..., a n }), n 2, s a K-system f any par {A, A j } ({a, a j }) for j s a K-par. Fnte K-systems have the followng property easly derved from the defnton: Proposton 1 (GS[6]). Let {A 1,..., A n } be a fnte K-system. For arbtrary dsjont subsets R 1 and R 2 of {1,..., n}, the par { R 1 A, R 2 A } s a K- par. Furthermore for every R 1, R 2 {1,..., n} we have A e A R 1 R 2 R 1 R 2 In order to be able to prove a property analogous to the above proposton, n the nfnte case we ntroduce a further unformty condton. We say that a system of sets {A < ω} s a unform K-system, f [A e ] and there s a computable functon r, such that for each j A A j W r(,j), A A j W r(,j),.e. {A, A j } s a K-par va the c.e. set W r(,j). The followng property holds for unform K-systems: Proposton 2 (GS[6]). Let {A < ω} be a unform K-system. Then for arbtrary dsjont computable sets R 1 and R 2, { R 1 A, R 2 A } s a K-par. Furthermore for every computable R 1 and R 2 A e A R 1 R 2 R 1 R 2 Proposton 1 and Proposton 2 show that every K-system of sets, ether fnte or unform, s a system of ncomparable reals. Thus whenever a K-system of degrees s bounded by a low degree t can be encoded by three parameters va the formula SW.

5 INTERPRETING TRUE ARITHMETIC IN G e 5 Exstence of fnte K-systems consstng of three sets s proven by Kalmulln [9]. The exstence of unform K-system (and thus n partcular of fnte K-systems of arbtrary cardnalty) s proven n [6]. Proposton 3. Let B be a non c.e. 0 2 set. Then there s a unform K-system {A < ω}, such that <ω A e B. 3. Codng sets and relatons n G e In ths secton we follow the lnes of Slaman and Woodn s [16] codng of countable sets and relatons n D e. The codng of arbtrary countable relatons n D e reles on the followng two assertons: () There s a formula codng every countable antchan va parameters. () For every set A, there s a set C = {C < ω} of ncomparable reals, such that for every X, Y e A and every, j < ω (3.1) C X e C j Y = j & X e Y. In order to prove a codng lemma for the local theory we shall need propertes analogous to () and (). The analogue of () s provded by the local verson of the Slaman-Woodn antchans codng theorem (Theorem 1). However, ths theorem s not as powerful as the global one. Indeed, t guarantees that a set of Σ 0 2 ncomparable reals s defnable by parameters n G e only n the case when t s unformly reducble to a low set. Thus we need to prove a stronger verson of (), namely we need to requre that the set C s unformly reducble to a low set. In the global theory property (3.1) s satsfed by every countable set C of reals that are mutually Cohen generc wth respect to meetng all dense sets that are arthmetc n A. Due to the genercty of ts elements, C s not bounded by A (n) for any n and hence t s not usable n the local theory. If we relax the condton of genercty only to the dense sets that are necessary to meet the property (3.1), we would obtan (by means of the usual forcng argument) an antchan C, for whch the best estmated upper bound s the frst jump of A whch s obvously not low. However t turns out that n G e we can use unform K-systems nstead of generc reals. Lemma 1. Let {A, B} be a nontrval K-par and let C = {C < ω} be a unform K-system bounded by B. Then for every X, Y e A and every, j < ω C X e C j Y = j & X e Y. Proof. Suppose that X, Y e A and that C X e C j Y. The second nequalty mples C e C j Y. On the other hand C e C Y and hence f t was the case j, {C, C j } would be a K-par and we would have C e Y e A. But C e A, so that = j. Thus C X e C Y. In partcular X e C Y. On the other hand X e X Y. But X e A and C e B so that {X, C } s a K-par. Hence X e Y. Lemma 1 shows that K-systems are antchans satsfyng (3.1) and thus approprate for codng n G e sets and relatons bounded by a half of a nontrval K-par. We wll prove that there s a suffcently large class of sets and relatons defnable va parameters n G e.

6 6 H. GANCHEV AND M. SOSKOVA Defnton 1. Let a G e and let R be a k-ary relaton n the nterval [0 e, a]. We shall say that R s e-presentable beneath a f there s an A a and a c.e. set W, such that R = {(d e (W 1 (A)), d e (W 2 (A)),..., d e (W k (A))) 1, 2,..., k W }. In partcular we shall say that R s e-presentable beneath a va A and W Let R be an n-ary relaton on degrees. For 1 k n let us denote by R(k) the k-th projecton of R,.e. R(k) = {r r 1,..., r k 1, r k+1,..., r k [(r 1,..., r k 1, r, r k+1,..., r k ) R]}. Note that f R s bounded by a,.e. the doman of R s bounded by a, then R(k) s also bounded by a. Furthermore f R s e-presentable beneath a va A and W, then R(k) s e-presentable beneath a va A and W (k), where W (k) = { 1,..., k 1, k+1,..., k [ 1,..., k 1,, k+1,..., k W ]}. Theorem 2. For every n 1 there s a formula ϕ n (x 1,..., x n, a, b, p 1,..., p 4n+2 ), such that for every half of a nontrval K-par a, and e-presentable beneath a n-ary relaton R, there are parameters b, p 1,..., p 4n+2, such that (x 1,..., x n ) R G e = ϕ n (x 1,..., x n, a, b, p 1,..., p 4n+2 ) Proof. Let a be a half of K-par and let a 1 be such that {a, a 1 } s a K-par. Take a K-par {b, b 1 } beneath a 1. Then {a, b, b 1 } s a K-system and n partcular {a, b} s a K-par. Note that snce {a b, b 1 } s also a K-par, a b s low. Fx an nteger n 1. For an arbtrary N denote by Dv() the dvsor and by Rem() the remander (or resdue) resultng from the dvson of by n. Let R be an n-ary e-presentable relaton beneath a va A and W. Fx B b and a unform K-system C = {C < ω} beneath B. For 1 k n denote by C(k) the unform K-system C(k) = {C Rem() = k}. Note that C(k) s unformly beneath B and hence the set C(k) = {d e (C) C C(k)} s defnable wth parameters. From ths defnton we obtan the parameters p 1,..., p 2n. We shall use C(k) to code the projectons R(k). Let C(k) + R(k) = {C W j (A) Dv() = j W (k) & Rem() = k} for 1 k n. Clearly C(k) + R(k) s unformly reducble to A B. Furthermore, Lemma 1 yelds that C(k)+R(k) s a set of ncomparable reals. Thus C(k)+R(k) = {d e (Y ) Y C(k) + R(k)} s unformly defnable va parameters n G e. Ths gves us parameters p 2n+1... p 4n. Besdes Lemma 1 yelds x a [ x R(k) c C(k)(c x C(k) + R(k)) ], and hence each of the projectons R(k) s unformly defnable va parameters. In order to code the relaton R we shall need one more antchan. Consder the set C W = {C 1 C n Rem( k ) = k, for 1 k n and Dv( 1 ),..., Dv( n ) W }. We clam that C W s a system of ncomparable reals unformly beneath B. Indeed, suppose that C 1 C n e C j1 C jn for some C 1 C n C W and C j1 C jn W. Hence, accordng to Proposton 2, { 1,..., n } {j 1,..., j n }. On the other hand Rem( 1 ) = Rem(j 1 ) = 1,..., Rem( n ) = Rem(j n ) = n, so that 1 = j 1,..., n = j n. Thus the set C W = {d e (C) C C W } s an antchan unform n b and so s defnable wth parameters b, p 4n+1 and p 4n+2.

7 INTERPRETING TRUE ARITHMETIC IN G e 7 Fnally let X 1,..., X n e A be such that there s a C 1 C n C W, such that for each 1 k n, C k X k s enumeraton equvalent to some C jk W Dv(jk )(A) C(k) R(k). Then Lemma 1 yelds k = j k and X k e W Dv(k )(A) for 1 k n. But Dv( 1 ),..., Dv( n ) W and hence (d e (X 1 ),..., d e (X n )) R. Thus for arbtrary x 1,..., x n a, the n-tuple (x 1,..., x n ) s an element of the relaton R f and only f c 1 C(1)... c n C(n)[ 1 k n(c k x k C(k)+R(k))] & c 1 c n C W ]. 4. Interpretng true arthmetc n G e Let us fx a fnte axomatzaton P A of arthmetc n the language {+,, <}, such that every model of P A has a standard part. Let B P A be the class of all models of P A. For arbtrary N B P A and x N let us denote by L N (x) the set of all elements of N less or equal to x n N,.e. (4.1) L N (x) = {z N N = z x} We have the followng charactersaton of the standard models of P A n B P A. Lemma 2. Let A B P A be a class of models of P A, contanng a standard model of arthmetc. Then for arbtrary N 1 B P A, N 1 s a standard model of arthmetc, f and only f for every x 1 N 1 and every N 2 A, there s an x 2 N 2 such that the sets L N1 (x 1 ) and L N2 (x 2 ) have the same cardnalty. Proof. Snce every model of P A has a standard part, N 1 s a standard model of arthmetc f and only f the set L N1 (x) s fnte for every x 1 N 1. Suppose that N 1 B P A s a standard model of arthmetc and let x 1 N 1. Then the set L N1 (x 1 ) has fnte cardnalty, say n. Take an arbtrary N 2 A. Snce N 2 has a standard part, then there s an x 2 N 2, such that L N2 (x 2 ) has cardnalty n. For the converse drecton suppose that N 1 B P A s such that for every x 1 N 1 and every N 2 A, there s x 2 N 2, such that the sets L N1 (x 1 ) and L N2 (x 2 ) have the same cardnalty. Snce there s a standard model of arthmetc n A, then the set L N1 (x 1 ) s fnte for arbtrary x 1 N 1 and hence N 1 s also standard. Fx a formula θ P A expressng the followng facts for an arbtrary degree a and parameters b N, p N, b +, p +, b, p, b < and p < (we shall denote such a lst of parameters by q): The relatons R + = {(x, y, z) G e = ϕ 3 (x, y, z, a, b +, p + )} and R = {(x, y, z) G e = ϕ 3 (x, y, z, a, b, p )} are (the graphs of two) bnary operatons on R N = {x G e = ϕ 1 (x, a, b N, p N )}. The relaton R < = {(x, y) G e = ϕ 2 (x, y, a, b <, p < )} s a bnary relaton on R N. (R N ; R +, R, R < ) s a model of P A. We shall denote ths model by N(a, q). Thus for every a G e the formula θ P A defnes a class N (a) of models of P A bounded by a. Now suppose that a s a half of a nontrval K-par. We clam that the class N (a) s nonempty and contans a standard model of arthmetc. Indeed, fx an ndependent system of reals X = {X < ω} unform n A a and denote by

8 8 H. GANCHEV AND M. SOSKOVA x the degree of X. Let R X N = {x < ω}, R X + = {(x, x j, x +j ), j < ω}, R X = {(x, x j, x j ), j < ω} and R X < = {(x, x j ) < j < ω}. It s clear that these relatons are e-presentable beneath a and hence they are defnable va parameters. Furthermore (R X N ; RX +, R X, R X <) s a standard model of arthmetc. Let N 1 and N 2 be two models n N (a) coded by q 1, and q 2 respectvely. For any x 1 N 1 and x 2 N 2 we shall say that (x 1, q 1 ) (x 2, q 2 ) f and only f there are parameters b and p for the formula ϕ 2, such that the relaton R = {(z 1, z 2 ) G e = (z 1, z 2, a, b, p ) s the graph of a bjecton from L N1 (x 1 ) nto L N2 (x 2 ) (n partcular the sets L N1 (x 1 ) and L N2 (x 2 ) have the same cardnalty). Note that f x 1 and x 2 represent the same standard natural number then (x 1, q 1 ) (x 2, q 2 ). Indeed, n ths case the sets L N1 (x 1 ) and L N2 (x 2 ) have the same fnte cardnalty, so that the set L N1 (x 1 ) L N2 (x 2 ) s fnte and hence e-presentable beneath a. Therefore every bjecton from L N1 (x 1 ) onto L N2 (x 2 ) s e-presentable beneath a and hence defnable by the formula ϕ 2. Thus (x 1, q 1 ) (x 2, q 2 ). Now for every half of a nontrval K-par a we can select the standard models of arthmetc n N (a) n the followng way: A model N 1 N (a) coded va the parameters q 1 s a standard model of arthmetc f and only f for every x N 1 and every N 2 N (a) (coded by, say, q 2 ), there s a y N 2, such that (x, q 1 ) (y, q 2 ). Thus we have proven the followng theorem. Theorem 3. There s a formula SMA such that for every half a of a nontrval K-par the followng assertons hold: () For every choce of parameters q, f G e = SMA(a, q), then N(a, q) s a standard model of arthmetc. () There are parameters q, such that G e = SMA(a, q). Thus we have defned a class of standard models of arthmetc n G e, namely N Ge = {N(a, q) a s a half of a K-par and G e = SMA(a, q)}. Hence for every arthmetcal sentence θ, N = θ f and only f for every K-par {a, b} there are parameters q, such that G e = SMA(a, q) and N (a, q) = θ. From here we obtan the computable translaton of the arthmetcal sentences nto sentences of G e. The results so far can be extended to a defnton of a parameterless standard model of arthmetc n G e. In order to do ths, t s enough to show that the equvalence relaton on tuples of the form (x, a, q), where a s a half of a K-par, G e = SMA(a, q) and x N(a, q), defned by (x 1, a 1, q 1 ) (x 2, a 2, q 2 ) x 1 and x 2 represent the same natural number, s defnable wthout parameters n G e. Frst of all note that f a 1, a 2 a for some half of K-par a we have (x 1, a 1, q 1 ) (x 2, a 2, q 2 ) (x 1, q 1 ) (x 2, q 2 ), for arbtrary q 1 and q 2, satsfyng G e = SMA(a 1, q 1 ) and G e = SMA(a 2, q 2 ). Thus we can compare all standard models bounded by a fxed half of K-par. We extend ths result by means of the followng lemma. Lemma 3. Let A 0 and A 1 be non c.e. 0 2 sets. Then there s a K-par {B 0, B 1 } such that B 0 e A 0 and B 1 e A 1.

9 INTERPRETING TRUE ARITHMETIC IN G e 9 Proof. Kallmuln [8] has proved the statement n the case A 0 = A 1. Snce the proof n the case A 0 A 1 s analogous, we gve here only a sketch. Fx 0 2 good approxmatons {A s 0} and {A s 1} of A 0 and A 1 respectvely. Defne by nducton on s the followng computable sequences V0 s and V1 s of fnte sets: Set V 0 0 = V 0 1 =. Suppose that V s 0 and V s 1 are defned. If V s (As ) \ V s (As+1 set ˆV s = V s ) = for 1,. Otherwse let k be the least natural number for whch there s ) =. Set ˆV s = V s an x and an 1, such that k, x V s(as ) \ V s and (As+1 ˆV s 1 = V s 1 { r, y, r k & r, y < s}. For each e, s and 1, denote by l(, e, s) the length of agreement between V s(as ) and W e s (here {We s } s a fxed c.e. approxmaton of W e ). For each 1 choose the least e such that l(, e, s) > max{l(, e, k) k < s}. If such an e does not exst set V s+1 V s+1 = ˆV s = ˆV s { e, y, {y} e, y < s}.. Otherwse set It s clear that the sets V = V s for 1 are c.e. We set B = V (A ). Now from the constructon of V 0 and V 1 t follows that B 0 and B 1 are not c.e. Furthermore the sequences {V0 s (A s 0)} and {V1 s (A s 1)} are 0 2 approxmatons to B 0 and B 1 respectvely, havng the followng property for arbtrary and s: ( V s (A s ) \ V s+1 (A s+1 ) ) ω [k] ω [ k] s V 1 (A 1 ). Kalmulln [8] has proved that the above property s a suffcent condton for {V 0 (A 0 ), V 1 (A 1 )} to be a K-par. Now let us turn to the proof of the defnablty n G e of the relaton. Let a 1 and a 2 be arbtrary halves of K-pars and let q 1 and q 2 be such that G e = SMA(a 1, q 1 ) and G e = SMA(a 2, q 2 ). Let x 1 N(a 1, q 1 ) and x 2 N(a 2, q 2 ) represent the same natural number. Fx a K-par {b 1, b 2 } such that b 1 a 1, b 2 a 2 and b 1 b 2 s half of a K-par (we can obtan such b 1 and b 2 applyng Lemma 3 and the trck used n the proof of Theorem 2). Let q 11 and q 22 be such that G e = SMA(b 1, q 11 ) and G e = SMA(b 2, q 22 ) and let x 11 N(b 1, q 11 ) and x 22 N(b 2, q 22 ) represent the same natural number as x 1 and x 2. Snce b 1 b 2 s half of a K-par, (x 11, q 11 ) (x 22, q 22 ). On the other hand b 1 a 1 = a 1 and a 1 s a half of a K-par so that (x 1, q 1 ) (x 11, q 11 ). Analogously (x 22, q 22 ) (x 2, q 2 ). Thus we obtan that (x 1, a 1, q 1 ) (x 2, a 2, q 2 ) f and only f there are b 1, b 2, q 11, q 22, x 11 and x 22 such that () {b 1, b 2 } s a K-par, b 1 a 1, b 2 a 2 and b 1 b 2 s half of a K-par. () G e = SMA(b 1, q 11 ) and G e = SMA(b 1, q 11 ). () x 11 N(b 1, q 11 ) and x 22 N(b 2, q 22 ) (v) (x 1, q 1 ) (x 11, q 11 ), (x 11, q 11 ) (x 22, q 22 ) and (x 22, q 22 ) (x 2, q 2 ). Thus the relaton s defnable n G e and we can buld a parameterless standard model of arthmetc. References [1] S. B. Cooper, Partal degrees and the densty problem. Part 2: The enumeraton degrees of the Σ 2 sets are dense, J. Symbolc Logc 49 (1984),

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