MAXIMAL CHAINS IN THE TURING DEGREES

Transcription

1 MAXIMAL CHAINS IN THE TURING DEGREES C. T. CHONG AND LIANG YU Abstract. W study th problm of xistnc of maximal chains in th Turing dgrs. W show that:. ZF + DC+ Thr xists no maximal chain in th Turing dgrs is quiconsistnt with ZF C+ Thr xists an inaccssibl cardinal ; 2. For all a 2 ω, (ω ) L[a] = ω if and only if thr xists a Π [a] maximal chain in th Turing dgrs. As a corollary, ZF C+ Thr xists an inaccssibl cardinal is quiconsistnt with ZF C+ Thr is no (bold fac) Π maximal chain of Turing dgrs.. Introduction. A chain in th Turing dgrs is a st of rals in which any two distinct lmnts ar Turing comparabl but not quivalnt. A maximal chain is a chain which cannot b proprly xtndd. An antichain of Turing dgrs, by contrast, is a st of rals in which any two distinct lmnts ar Turing incomparabl. A maximal antichain is an antichain that cannot b proprly xtndd. In this papr, w study maximal chains in th Turing dgrs. This is a classical topic in rcursion thory which may b tracd back to Sacks [3], in which h provd th xistnc of a minimal uppr bound for any countabl st of Turing dgrs. As a consqunc, assuming th Axiom of Choic AC, thr is a maximal chain of ordr typ ω. Abraham and Shor [] vn constructd an initial sgmnt of th Turing dgrs of ordr typ ω. All of ths rsults dpnd havily on AC. W ar intrstd in th following qustions:. Is AC ncssary to show that thr xists a maximal chain? Can on construct a maximal chain without AC? 2. Is thr a dfinabl, say Π, maximal chain? For (), w will prov in Sction 2 that ovr ZF plus th Axiom of Dpndnt Choic DC, thr is no maximal chain in th Turing dgrs is quiconsistnt with ZF C + thr xists an inaccssibl cardinal. This shows that th xistnc of maximal chains is dcidd by on s blif, ovr ZF + DC, in 2000 Mathmatics Subjct Classification. 03D28,03E5,03E35,03E45. Th authors wish to thank th rfr for a carful rading of th manuscript and for suggstions that improvd on th original rsults and ld to Thorm 3.5 and Corollary 3.6. W also thank Manul Lrman, Richard Shor and Yu Yang for hlpful discussions. Th first author wishs to thank Andra Sorbi and th Univrsity of Sinna for thir gracious hospitality, during a visit undr th INDAM-GNSAGA visiting profssorship schm. His rsarch was also partially supportd by NUS grant WBS Th scond author was supportd by NUS Grant No. R (Singapor) and NSF of China No and No

2 2 C. T. CHONG AND LIANG YU th xistnc of an inaccssibl cardinal. Thus if on blivs that thr is no larg cardinal in L, thn on would dduc th xistnc of a maximal chain from ZF + DC. On th othr hand, if thr is no maximal chain thn ω is inaccssibl in L (in fact in L[a] for any ral a). For (2), th situation is mor complicatd. It is not difficult to show, undr ZF + DC, that thr xists no Σ maximal chain in th Turing dgrs. This is th bst rsult within ZF + DC. By Martin and Solovay s rsult [9], it is consistnt with ZF C that vry maximal chain is Π. Is it consistnt with ZF C that thr xists a Π maximal chain? W show (Thorm 3.5) that thr is a Π maximal chain undr th assumption of (ω ) L = ω. W organiz th papr as follows: In Sction 2, w study th rlation btwn xistnc of maximal chains and larg cardinals. In Sction 3, w considr th problm of th xistnc of dfinabl maximal chains. Notations. A tr T is a subst of 2 <ω which is downward closd. Givn a tr T, a finit string σ is said to b a splitting nod on T if both σ 0 and σ ar in T. Th string σ is said to b an n-th splitting nod on T if σ T is a splitting nod and thr ar n -many splitting nods that ar initial sgmnts of σ. Lt T σ = {τ T τ σ τ σ} and T n = {σ T σ n}. W us [T ] to dnot th st of rals {z ( n)(z n T )}. A prfct tr T is a tr in which for ach σ, thr is a τ T so that τ σ and τ is a splitting nod. Dfin th n-th lvl of T to b Lv n (T ) = {σ Lv n (T ) ( τ)(τ T σ τ τ is an n-th slpitting nod on T )}. For notations and dfinitions not givn hr, s [4], [8], [6],[2] and [4]. Acknowldgmnt. W thank th rfr for a carful rading of th manuscript and for suggstions that improvd on th original rsults and ld to Thorm 3.5 and Corollary 3.6. W also thank Manul Lrman, Richard Shor and Yu Yang for hlpful discussions. 2. Chains and larg cardinals. W us I to dnot th statmnt thr xists an inaccssibl cardinal and N M T to dnot thr xists no maximal chain in th Turing dgrs. In this sction, w prov that ZF + DC + NMT is quiconsistnt with ZF C + I. A st A of rals is said to b an antichain if for all x, y A, x y implis x T y. Th following rsult is ssntially du to Sacks [3] which will play a critical rol in this and latr sctions. Lmma 2.. (ZF + DC) For ach prfct tr T, thr is a prfct tr S T so that [S] is an antichain. Proposition 2.2. Assum Con(ZF C + I). Thn Con(ZF + DC + NMT ) holds.

3 MAXIMAL CHAINS IN THE TURING DEGREES 3 Proof. By Solovay s thorm [7], assuming Con(ZF C + I), ZF + DC is consistnt with th statmnt: Evry uncountabl st contains a prfct subst. Now vry maximal chain is uncountabl and, by Lmma 2., dos not contain a prfct st. Hnc ZF + DC is consistnt with th statmnt Thr is no maximal chain in th Turing dgrs. Proposition 2.3. If Con(ZF + DC + NMT ), thn Con(ZF C + I). Proof. By Solovay s rsult [6], it suffics to prov that ZF + DC + NMT ( x 2 ω )(ω L[x] < ω ). To show this, assum that thr is a ral x so that ω L[x] = ω. Thn sinc 2 ω L[x] has a wll-ordring in L[x] of ordr typ ω L[x], it is straightforward to obtain a maximal chain A L[x] 2 ω in L[x] of siz (ℵ ) L[x]. Sinc ω L[x] = ω, A = ℵ. Lt A = {z α } α<ω. If A is not a maximal chain in V (th ral world), thn tak a witnss z A which is comparabl with all of th rals in A. So thr is an α < ω so that z T z α. Thn z L[x] sinc A L[x]. So A is not a maximal chain in L[x], a contradiction. Not that th proofs abov also show that ZF C + I is quiconsistnt with ZF + DC+ Evry chain of th Turing dgrs is countabl. 3. Th xistnc of dfinabl maximal chains. In this sction, w study th xistnc of dfinabl maximal chains of Turing dgrs. Proposition 3.. (ZF + DC) Thr is no Σ maximal chain in th Turing dgrs. Proof. If A is a maximal Σ chain in th Turing dgrs, thn A must hav a prfct subst sinc A is uncountabl. By Lmma 2., A will thn contain a pair of T -incomparabl rals, which contradicts th fact that A is chain. Assuming MA + CH + (ω ) L = ω, by Martin-Solovay s rsult that ach st of rals with siz at most ℵ is Π [9], on ss that ach maximal chain in th Turing dgrs is a Π -st. Th qustion thn is whthr thr is a Π maximal chain. W giv a positiv answr assuming (ω ) L = ω. Rcall that a ral x is a minimal covr of a countabl st A of rals if (i) x is an uppr bound of vry ral in A, and (ii) no y < T x is an uppr bound of A. Sacks [3] showd that vry countabl collction of Turing dgrs has a minimal covr. Th nxt lmma implis that a minimal covr xists with arbitrarily high doubl jump. Lmma 3.2. (ZF ) Assum A is a countabl st of rals and x is a ral. Thr is a minimal covr z of A so that z T x.

4 4 C. T. CHONG AND LIANG YU Proof. W fix an ffctiv numration of partial rcursiv oracl functions {Φ } <ω. Lt {p n } n<ω b a rcursiv - numration of prim numbrs. Allowing ambiguity, w will also us p σ to dnot th prim numbr which cods th string σ. Givn a rcursiv oracl function Φ, w us Φ y = T to xprss th following:. Φ y is total; 2. ( n)(φ y (n) = σ Lv n(t ) p σ). Dfin L n (T ) to b th lftmost n + -th splitting nod and R n (T ) to b th rightmost n + -th splitting nod. For ach finit string σ, ral x and prfct tr T 2 <ω with σ T, dfin T (σ, x, T ) to b a prfct tr so that T (σ, x, T ) is th intrsction of a squnc of prfct trs T n givn as follows: T 0 = T σ. T n+ = {τ σ Lv n (T n )(τ σ)} S n+ T n whr S n+ is dfind as: Cas() : x(n) = 0. σ S n+ if and only if thr xists τ Lv n+ (T n ) so that σ τ and τ = L (Tn ν ) for som ν which is an n-th splitting nod of T n. Cas(2) :x(n) =. σ S n+ if and only if thr xists τ Lv n+ (T n ) so that σ τ and τ = R (Tn ν ) for som ν which is an n-th splitting nod of T n. Dfin T (σ, x, T ) = T n. n In othr words, T (σ, x, T ) is a subtr which, roughly spaking, cods x(n) at a 2n-th splitting nod of T σ. Not that T (σ, x, T ) T T x. Morovr, suppos for som rcursiv oracl function Φ, w hav Φ y = T for all y [T ]. Thn thr is a rcursiv oracl function Ψ such that Ψ y = x for all y T (σ, x, T ). Furthrmor, givn an indx of th oracl function Φ, an indx of th oracl function Ψ may b ffctivly obtaind from σ. In othr words, thr is a rcursiv function f such that Φ y f(σ ) = x for all y [T (σ, x, T )]. Sinc Φ y = T for all y [T (σ, x, T )] [T ] and x T T T (σ, x, T ), thr is a rcursiv function g so that Φ y g(σ ) = T (σ, x, T ) for all y [T (σ, x, T )]. W giv a sktch of th ida bhind th construction of z. To obtain a minimal covr of a countabl st A = {x i } i<ω, on maks appropriat modifications of th construction of a minimal dgr (s [8]). To mak th minimal dgr rlativly high (i.. to mak it comput a givn x through jumps), on nds to cod th indics of th prfct trs in th cours of th construction ([5] is a good sourc whr this ida is mad prcis). Wr th construction uniform, on could us th Rcursion Thorm to cod th indx of th nxt prfct tr bing dfind during th stp by stp construction. This tchniqu could b applid to cod x and x i into z for ach i < ω. Howvr, to achiv minimality, th construction is non-uniform (on nds to dcid whthr th nxt tr will b an -splitting tr or a full tr, which in gnral is a doubl jump qustion). Although it is highly non-uniform, th construction dos bcom uniform onc it is dcidd which situation on is in (s Substp 3 of th construction blow). This is th rason for using z to gt up to x. W now turn to th construction.

5 MAXIMAL CHAINS IN THE TURING DEGREES 5 Fix a ral x and an numration {x i } i ω of A. Suppos th rcursiv oracl functional Φ 0 satisfis Φ y 0 = 2<ω for all rals y. W construct a squnc of prfct trs stp by stp. At stp n, w construct a prfct tr T n T i<n x i and a finit string σ n so that T n σ n = {τ τ σ n }, σ n > n and thr is a rcursiv oracl functional Φ n so that for ach y [T n ], Φ y n = T n. Construction At stp 0, dfin T 0 = 2 <ω and σ 0 =. At stp n +, thr ar thr substps: Substp (Coding x). For ach k: If x(n) = 0, dfin σn+,0 k = R (T L k(t n ) n ); othrwis, dfin σn+,0 k = L (T R k(t n ) n ). Dfin T n+,0,k = T σk n+,0 n. Hnc thr is a rcursiv oracl functional Φ ik such that for ach y T n+,0,k, Φ y i k = T n+,0,k. Not that th function k i k is rcursiv. It follows from th Rcursion Thorm that thr is a numbr k 0 such that Φ y i k0 = Φ y k 0 for all y [T σk 0 n+,0 n ]. Fix this k 0 and dfin σn+ 0 = σ k 0 n+,0, and T n+,0 = T n+,0,k0. Substp 2 (Coding x n ). For ach k: Dfin T n+,,k = T (R (T L k(t n+,0 ) n+,0 ), x n, T n+,0 ). By th discussion abov, thr ar rcursiv functions f, g so that Φ y f(k) = x n and Φ y g(k) = T n+,,k for all y [T n+,,k ]. By th Rcursion Thorm, thr is a k such that Φ y k = Φ y g(k ) = T n+,,k for all y [T n+,,k ]. Dfin σ n+, = R (T L k (T n+,0) n+,0 ) and T n+, = T n+,,k. Substp 3 (Forcing a minimal covr). This is th only plac whr z is usd. Cas() : Thr xists σ T n+, and a numbr i σ so that for all m > i σ, τ, τ 2 σ, {τ, τ 2 } T n+, Φ τ n+ (m) Φτ 2 n+ (m) = Φτ n+ (m) = Φ τ 2 n+ (m). Choos th last such σ T n+, (in th sns of th coding of strings). Dfin S = Tn+,. σ For ach k, dfin S k = S R (S L k (S)). By th Rcursion Thorm again, thr is a k 2 so that Φ y k 2 = S R (S L k 2 (S) ) for all y [S R(SL k (S) 2 ) ]. Dfin σ n+ = R (S L k (S) 2 ) and T n+ = S σn+. Lk(Tn+,) R(Tn+, ) Cas(2) : Othrwis. For ach k, dfin S k = Tn+,. Thn w can S k -rcursivly find a sub-prfct tr P k of S k such that for all y [P k ], Φ y n is total and for all τ 0, τ P k, if τ 0 τ, thn thr must b som i so that for all rals z 0 τ 0 and z τ, Φ z0 n (i) Φ z n (i). By th Rcursion Thorm, thr is a k 2 such that Φ y k 2 = P k2 for all y [P k2 ]. Lt T n+ = P k2 and σ n+ = R (T L k 2 (T n+,) n+, ) T n+. Lt n+ = k 3. Thn Φ y n+ = T n+ for all y [T n+ ]. This complts th construction at stp n +. Not that by induction on n, T n+ T i<n+ x i. Finally, lt z = n σ n. By th usual argumnts (s [8]), on can show that z is a minimal covr of A sinc T n+ T i<n+ x i for ach n. W show that z T x. W prov this by

6 6 C. T. CHONG AND LIANG YU using induction on n to show that x(n) may b z -uniformly computd. To do this, w show that th indx n is uniformly computd from z. At stp n +, by inductiv hypothsis, w can z -rcursivly find n. By th construction, w hav Φ y n = T n for all y [T n ]. W can z-rcursivly find an initial sgmnt of z which is R (T L k 0 (T n ) n ) or L (T R k 0 (T n ) n ) for som k 0. In th first cas, x(n) = 0. In th scond cas, x(n) =. Not that Φ z k 0 is th prfct tr in Substp of th construction. W can z-rcursivly find k so that R ((Φ z k 0 ) L k (Φ z k ) 0 ) z. Thn Φ z k = T (R ((Φ z k 0 ) L k (Φ z k ) 0 ), x n, Φ z k 0 ) = T n+,. W us z to dcid which cas Φ z k is in. In cas (), w z -rcursivly find th last σ T n+, with th rquird proprty. Lt S = Tn+,. σ Thn w z-rcursivly find th k 2 so that Φ z k 2 = S R(SL k 2 (S) ). So n+ = k 2 and T n+ = S R (S L k (S) 2 ) = Φ z k 2. In Cas (2), w z-rcursivly find th k 2 so that R (T L k 2 (T n+, ) n+, ) z. Thn by th construction, n+ = k 2 and T n+ = Φ z k 2. Thus z T x and th proof of Lmma 3.2 is complt. Rmark. W conjctur that z may b rplacd by z in Lmma 3.2. Corollary 3.3. (ZF +DC) Assum that A is a countabl st of rals. Thr is a ral x so that for ach y T x, thr is a minimal covr z of A so that z T y. Proof. Assum A is a countabl st of rals. Fix an numration {x i } i of A. Th st B = {y ( z)(z is a minimal covr of A z T y)} = {y ( )( i)( j)[(φ y is total (Φ y ) T y x i T Φ y (Φ Φy j is total ( k)(x k T Φ Φy j ) = Φ y T Φ Φy j )]} is a Borl st. By Lmma 3.2, for ach ral x, thr is a ral y B so that y T x. By Borl dtrminacy (Martin [0]), thr is a ral x so that {y y T x} B. To show th main rsult, w first construct a Π maximal chain in th Turing dgrs undr th assumption V = L. Th proof of th following lmma dpnds havily on th rsults of Boolos and Putnam [3]. Call a st E ω ω an arithmtical copy of a structur (S, ) if thr is a - function f : S ω so that for all x, y S, x y if and only if (f(x), f(y)) E. In ([3]) it is provd that if (L α+ \ L α ) 2 ω thn thr is an arithmtical copy E α L α+ of (L α, ) so that any x (L α+ \ L α ) 2 ω is arithmtical in E α (i.. E α is a mastr cod for α in th sns of Jnsn [7]). Morovr, ach z L α 2 ω is on-on rducibl to E α. Sinc E α ω ω, it may b viwd as a ral. Not that for ach constructibly countabl β, thr is an α > β such that (L α+ \ L α ) 2 ω. W will b considring sts A α ω. It will b convnint to idntify A with an α-squnc {A γ γ < α} of rals, whr A γ = {(γ, n) (γ, n) A}.

7 MAXIMAL CHAINS IN THE TURING DEGREES 7 Thorm 3.4. Assum V = L. Thr is a Π maximal chain in th Turing dgrs of ordr typ ω. Proof. By th Gandy-Spctor thorm, a st A of rals is Π if and only if thr is a Σ 0 -formula ϕ such that y A ( x L ω y [y])(l ω y [y] = ϕ(x, y)), whr ω y is th last ordinal α biggr than ω so that L α[y] is admissibl (s Thorm 3. Chaptr IV [2]). Our proof combins Corollary 3.3 and van Engln t al s argumnt [5]. Basd on th papr [5], Millr [] providd a gnral machinry to construct a Π st satisfying som particular proprtis. Howvr, th prsntation is sktchy and incomplt. W giv a dtaild argumnt hr whr it prtains to th thorm at hand. Assuming V = L, w dfin a function F on ω α<ω P(α ω) as follows: For ach α < ω, A α ω, w dfin F (α, A) to b th ral y such that thr xists a lxicographically last tripl (β, E, 0 ) (whr th ordring on th scond coordinat is < L ) satisfying th following proprtis :. Thr is a function h L β which maps ω onto α, and (L β+ \ L β ) 2 ω ; 2. E L β+ is an arithmtical copy of (L β, ) with th proprtis mntiond bfor th statmnt of Thorm 3.4; 3. y is a minimal covr of th st of rals {x β β < α n(x β (n) = A(β, n))}; 4. y T E and finally 5. y = Φ E 0. W show that F is a total function. For ach (α, A), w show that thr xists a y such that F (α, A) = y. It suffics to show that (β, E, 0 ) xists. Thn by th fact that th lxicographical ordr is a wll ordring, thr must b a last on and this will yild th y ndd to dfin F (α, A). Fix a ral x for A as in Corollary 3.3, th bas of a con of Turing dgrs that ar doubl jumps of minimal covrs of A. Sinc V = L, thr is a β > α such that x L β, (L β+ \ L β ) 2 ω and thr is a function h α mapping ω onto α. By th discussion abov, thr is an arithmtical copy E ω ω in L β+ such that E > T x. By Corollary 3.3 and th choic of x, thr is a minimal covr y of A so that y T E and y = Φ E 0 for som 0. Obviously, L β+ L ω y [y]. By th absolutnss of < L, it is asy to s that F (α, A) is dfind. Not that F (α, A) dpnds ssntially on A sinc A is a squnc of rals of lngth α. Morovr, for such A s on can vrify using th absolutnss of < L that thr is a Σ 0 formula ϕ(α, A, z, y) such that F (α, A) = y L (A,y) ω [A, y] = z h(ϕ(α, A, z, y) h is a function from ω onto α). Thus w can prform transfinit induction on countabl ordinals to construct a maximal chain of Turing dgrs of ordr typ ω. But car has to b xrcisd hr sinc in gnral sts constructd this way ar Σ ovr L ω, i.. Σ 2 and not ncssarily Π.

8 8 C. T. CHONG AND LIANG YU Dfin G(α) = y if and only if α < ω y f(f (2ω ) α+ f L ω y[y] f(α) = y β(β < α = f(β) = F (β, {(γ, n) n f(γ) γ < β}))). Sinc L ω y [y] is admissibl, G is Σ -dfinabl. In othr words, G(α) = y if and only if thr is a function f : α + 2 ω with f L ω y[y] such that L ω y[y] = (( s)( β α)( z s)ϕ(β, {(γ, n) n f(γ) γ < β}, z, f(β))) f(α) = y. Dfin th rang of G to b T. Thn y T if and only if thr xists an ordinal α < ω y and a function f : α + 2ω with f L ω y[y] such that L ω y[y] = (( s)( β α)( z s)ϕ(β, {(γ, n) n f(γ) γ < β}, z, f(β))) f(α) = y. So T is Π. All that rmains is to show that G is a wll-dfind total function on ω. This can b don using th sam argumnt as that for showing th rcursion thorm ovr admissibl structurs (s Barwis [2]). Th only difficult part is to argu, as was don arlir, that th function f dfind abov xists. W lav this to th radr. Thus T is a chain of ordr typ ω. To s that it is a maximal chain, lt x b a ral which is T-comparabl with all mmbrs of T. Slct th last α such that G(α) T x. Thn x T G(β) for all β < α. Sinc G(α) is a minimal covr of {G(β) β < α}, w hav G(α) T x. Thus T is a Π maximal chain. W arriv at th following charactrization: Thorm 3.5. Assum ZF + DC. Th following statmnts ar quivalnt:. (ω ) L = ω ; 2. Thr xists a Π maximal chain in th Turing dgrs. 3. Thr xists a Π uncountabl chain in th Turing dgrs. Proof. () = (2): Suppos (ω ) L = ω. Fix th Π st T as in Thorm 3.4. Sinc th statmnt T is a chain is Π 2 and L = T is a chain, T is a chain in th ral world V. Sinc T is uncountabl in L and (ω ) L = ω, T is uncountabl. Thus if x is a ral so that {x} T is a chain, thn x < T y for som y T so that x L. Sinc L = T is a maximal chain, T is a maximal chain in V. (2) = (3): This is Obvious. (3) = (): Suppos T is a Π uncountabl chain in th Turing dgrs. By Lmma 2., T is a thin st. Solovay [6] provd that if T is a thin Π st, thn T L, and (T ) L = T L = T. Thus T L (ω) L. Sinc T is uncountabl, (ω ) L = ω. Now Thorm 3.5 may b rlativizd to any ral a. To do this on first obsrvs that an analog of th Boolos-Putnam thorm [3] on arithmtic copis holds, so that if L α+ [a] \ L α [a], thn thr is an E α L α+ [a] 2 ω ω in which vry ral in L α+ [a] is a-arithmtical (i.. arithmtical in E α a). This provids th stting for stablishing a rlativizd vrsion of Thorm 3.4, namly if V = L[a],

9 MAXIMAL CHAINS IN THE TURING DEGREES 9 thn thr is a Π [a] maximal chain in th Turing dgrs of ordr typ ω. With this, on drivs a rlativizd vrsion of Thorm 3.5, whr ω L is rplacd by ω L[a], and Π by Π [a]. This lads to th following corollary showing that bold fac Π maximal chains play a critical rol in th xistnc problm of maximal chains, and givs an answr to th scond qustion posd at th bginning of this papr. Corollary 3.6. Th following statmnts ar quiconsistnt: () ZF C + I; (2) ZF + DC + Thr xists no Π maximal chain in th Turing dgrs. (3) ZF + AC + Thr xists no Π maximal chain in th Turing dgrs. Proof. () = (2). Assum that ZF C +I is consistnt. Thn by Proposition 2.2, ZF + DC + Thr xists no Π maximal chain in th Turing dgrs is consistnt. (2) = (). Assum that ZF + DC + Thr xists no Π maximal chain in th Turing dgrs is consistnt. By th obsrvation abov on rlativizing Thorm 3.5, th xistnc of a Π [a] maximal chain of Turing dgrs is quivalnt to ω L[a] = ω. Thus if thr is no Π maximal chain in th Turing dgrs, thn ω L[a] < ω for all rals a. This implis that ZF C + I is consistnt. (3) = (2). Obvious. () = (3). Assum that ZF C + I is consistnt. Thn ZF C + ω is inaccssibl in L is consistnt (by Lvy collaps). So thr is a ZF C modl M so that M = x 2 ω (ω L[x] < ω ). By th rlativizd vrsion of Thorm 3.5, thr is no Π maximal chain in th Turing dgrs in M. REFERENCES [] Uri Abraham and Richard A. Shor, Initial sgmnts of th dgrs of siz ℵ, Isral J. Math., vol. 53 (986), no., pp. 5. [2] Jon Barwis, Admissibl sts and structurs, Springr-Vrlag, Brlin, 975. [3] Gorg Boolos and Hilary Putnam, Dgrs of unsolvability of constructibl sts of intgrs, J. Symbolic Logic, vol. 33 (968), pp [4] Kith J. Dvlin, Constructibility, Prspctivs in Mathmatical Logic, Springr- Vrlag, Brlin, 984. [5] Fons van Engln, Arnold W. Millr, and John Stl, Rigid Borl sts and bttr quasi-ordr thory, Logic and combinatorics (arcata, calif., 985), Contmp. Math., vol. 65, Amr. Math. Soc., Providnc, RI, 987, pp [6] Thomas Jch, St thory, Springr Monographs in Mathmatics, Springr-Vrlag, Brlin, [7] R. Björn Jnsn, Th fin structur of th constructibl hirarchy, Ann. Math. Logic, vol. 4 (972), pp ; rratum, ibid. 4 (972), 443. [8] Manul Lrman, Dgrs of unsolvability, Prspctivs in Mathmatical Logic, Springr-Vrlag, Brlin, 983, Local and global thory. [9] D. A. Martin and R. M. Solovay, Intrnal Cohn xtnsions, Ann. Math. Logic, vol. 2 (970), no. 2, pp

10 0 C. T. CHONG AND LIANG YU [0] Donald A. Martin, Borl dtrminacy, Ann. of Math. (2), vol. 02 (975), no. 2, pp [] Arnold W. Millr, Infinit combinatorics and dfinability, Ann. Pur Appl. Logic, vol. 4 (989), no. 2, pp [2] Yiannis N. Moschovakis, Dscriptiv st thory, Studis in Logic and th Foundations of Mathmatics, vol. 00, North-Holland Publishing Co., Amstrdam, 980. [3] Grald E. Sacks, Dgrs of unsolvability, Princton Univrsity Prss, Princton, N.J., 963. [4], Highr rcursion thory, Prspctivs in Mathmatical Logic, Springr- Vrlag, Brlin, 990. [5] Stphn G. Simpson, Minimal covrs and hyprdgrs, Trans. Amr. Math. Soc., vol. 209 (975), pp [6] Robrt M. Solovay, On th cardinality of Σ 2 sts of rals, Foundations of mathmatics (symposium commmorating kurt gödl, columbus, ohio, 966), Springr, Nw York, 969, pp [7], A modl of st-thory in which vry st of rals is Lbsgu masurabl, Ann. of Math. (2), vol. 92 (970), pp. 56. DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE NATIONAL UNIVERSITY OF SINGAPORE LOWER KENT RIDGE ROAD SINGAPORE chongct@math.nus.ud.sg INSTITUTE OF MATHEMATICAL SCIENCE NANJING UNIVERSITY NANJING, JIANGSU PROVINCE P.R. OF CHINA yuliang.nju@gmail.com