Second order logic and first order theories of reducibility orderings*

J. Barwise, H. J. Keisler and K. Kunen, eds., The KIeene Symposium BNorth-Holland Publishing Company (1980) 181-200

Second order logic and first order theories of reducibility orderings* Anil Nerode and Richard A. Shore

Department of Mathematics, Cornell University, Ithaca, N.Y. 14853, U.S.A. Dedicated to Professor S. C, Kleene on the occasion of his 70th birthday Abstract: We show that the first order theories of many reducibility orderings are recursively isomorphic to second order logic on countable sets (and so to true second order arithmetic). The reduction procedure uses some initial segment results and Spector’s theorem on countable ideals in the degrees to code quantification over symmetric irreflexive binary relations. This is known to be enough to obtain full second order logic. Applications to other theories are mentioned as are several to problems of definability in, and automorphisms of, the Turing degrees.

0. Introduction

The first order theory of relational systems has been a mainstay of (1879). For many of the common relamathematical logic since FREGE tional systems arising in classical mathematics, the degrees of unsolvability of their complete theories are easy to calculate. They are usually decidable, or full first order arithmetic, or at least analytic in the sense of KLEENE (1955). It was, therefore, somewhat surprising when SIMPSON (1977) proved that the first order theory of the Turing degrees with just the ordering relation has the same degree as the full second order theory of true arithmetic. There were, however, some precedents for such results. In particular within recursion theory itself ELLENTUCK (1970), ( 1973) showed that the first order theory of the isols based on addition, multiplication (and weak exponentiation) was equivalent to second order arithmetic. In model theory a number of results have been derived by showing that the theory of the infinitely generic or existentially universal models of some theory T (e.g., division rings or groups) is equivalent to second order (1975), Ch. 16, arithmetic. (See, for example, HIRSCHFELD and WHEELER CHERLIN(1976), Ch. IV, Section 3.) All of these results were proved by finding special devices to first code in a countable set equipped with *The preparation of this paper was partially supported by NSF Grant MCS 77-04013. We would also like to thank C. Jockusch and M. Lerman for helpful conversations and correspondence.

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operations coding and X on the natural numbers and by then using some other device to code monadic second order quantification over this set. We will present a quite different and apparently simpler plan of attack on the general problem of coding second order arithmetic. The main idea is to avoid arithmetic entirely and go .directly back to general relational systems. That is we will code full second order logic by coding in quantification over arbitrary relations. The major simplification over coding arithmetic that this allows is based on the fact that it actually suffices to code all symmetric irreflexive binary (s.i.b.) relations. In the and setting of first order arithmetic itself this idea first appears in CHURCH QUINE(1952) who showed how to reduce first order arithmetic to one such et aL(l965,Theorem 3.3.3)) relation. Later L ~ v ~ 0 ~ ( 1 9 6 3 ) (also s e e ERSHOV and RABINand SCOTT(n.d.) showed in a general setting how all relations could be coded by s.i.b. ones. They then exploited this argument (RABIN, 1965) to give simple proofs of undecidability results for first order theories by coding such relations in their models. For an exposition of many such (1965, p. 58, and 1977, pp. 614-615), results see ERSOV(1965). UIN observes that analogous results hold for higher order languages as well but as he did not see any interesting applications, he omitted detailed consideration of second order problems. We will fill this gap by providing such a treatment with suitable applications. The point will be that, as all relations can be coded by s.i.b. ones, quantification over s.i.b. relations gives full second order logic. Although we will touch on other applications of these methods we will deal in detail only with the case of reducibility orderings. The key ingredient of our coding is the theorem of SPECTOR(1956) that every countable ideal I of degrees has an exact (i.e., minimal) pair of degrees (a, b) over it (i.e., I = {xlx < a&x < b}). We want to exploit it by devising a coding that converts quantification over arbitrary countable s.i.b. relations to quantification over ideals in the degrees. Spector’s theorem then allows us to convert this into first order quantification over pairs of elements thereby giving an interpretation of second order logic on countable structures in the first order theory of the degrees. In addition to avoiding the particularities of arithmetic a second type of simplication is achieved. Unlike SIMPSON(1977) we need no results using the jump operator nor even any new structural lemmas about the degrees. Indeed other than Spector’s theorem on ideals, we only need some old initial segment results. Lachlan’s theorem (LACHLAN,1968) that every countable distributive lattice can be embedded as an initial segment of the degrees is more than sufficient. It is fitting for this symposium that the structural facts we need about the degrees can be traced quite directly to the seminar in logic conducted

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by Professor Kleene in Madison in 1953. The proof of Spector's theorem and POST on ideals actually is essentially contained in Section 4 of UEENE (1956) (1954) on which the seminar was based. It first appears in SPECTOR which was a direct outgrowth of that seminar. Of course, this paper also contains the direct ancestor of all initial segment results: There is a minimal degree. This theorem answered a major open question in the Kleene and Post paper. As is frequently the case, the rewards of simplication are greater generality and wider applicability. Since our constructions rely on only these two properties of the Turing degrees we can almost immediately deduce the same result for many other reducibilities. Thus the first order theories of one-one, many-one, truth-table, weak-truth-table and arithmetic degrees as well as Turing degrees are all recursively isomorphic to true second order arithmetic (or equivalently second order logic on countable sets). Analogous results also hold for reasonable substructures of these orderings as well. The proofs, however, are a bit more technical and will be included in NERODEand SHORE(1980). We will, however, discuss these results and the key ingredients of their proofs along with many other applications in the fourth section of this paper. The applications will give much information about automorphisms of the Turing degrees and general definability questions for many reducibility orderings. For the first area, for example, we can show that every automorphism of the Turing degrees is the identity on a cone. As to definability in the Turing degrees we can show that every class closed under the jump which is definable in true second order arithmetic is definable in ('3, <,On) as is the a-jump and every such relation on degrees above 0'"'. Again these results will be discussed in Section 4 but the detailed computations and proofs will appear elsewhere. In this paper we take the liberty of proceding at a leisurely pace. As we view this paper as propaganda for a particular approach to coding second order logic or arithmetic as well as an expository work, we will go into more detail than might otherwise be necessary in the first two sections. Readers familiar with undecidability results in general should just skim these sections. The first section will consist of a review of material on codings using s.i.b. relations with enough pictures to convince the reader that quantification over such relations suffices for all of second order logic. In Section 2 we will show how to describe in a first order way some distributive lattices such that quantification over lattice ideals codes quantification over all s.i.b. relations on the atoms of the lattice. Thus full second order logic can be interpreted in the theory of distributive lattices with monadic quantification over ideals. It is then an easy matter to translate this into the first order theory of the Turing degrees using

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Spector's theorem on countable ideals and Lachlan's embeddmg theorem. This will be done in Section 3 along with similar treatments for the other reducibility orderings. The last section will, as we have mentioned, discuss applications and generalizations of these methods. 1. s i b . relations and second order logic

We will begin by essentially reproducing pictures and explanations from (n.d) (some of which appears in an unpublished paper of RABIN and SCOTT RABIN(1964)) that show how to code arbitrary relations by s.i.b. ones. We will then try to make explicit the use of these codings for interpreting full second order logic that is implicit in RABIN(1965). (It was intentionally omitted from that account and that of RABIN (1977, $3) (which we also recommend) as it was irrelevent to the concerns of those papers.) 1.1. Binary relations. Suppose we are given a structure % = ( A , R ) with R an arbitrary binary relation. We wish to construct % ' = ( A ' , S ) so that A ' > A and to code R by the symmetric irreflexive binary relation S on a subset of A' definable from S . Relativization will then supply an effective transformation Fl taking a formula 'p involving R to one 'pFl involving S such that % i = ' p o % ' k ' p F ~ . For each x E A we will add two new elements t l ( x ) and t 2 ( x ) tagging it as being in the field of R . For each pair ( x , y ) such that R ( x , y ) we also add three new elements u(x,y), u ( x , y ) and w ( x , y ) to form the set A'. The s.i.b. relation S is then defined as indicated by the connecting lines in Fig. 1.1. To be quite explicit we say that S ( x , t i ( x ) )holds for each x € A . Also, for each x,y such that R(x,y), all of S(x,u(x,y)), S(x,u(x,y)), S(w(x,y), u(x,y)), S(w(x,y),y)and their converses hold. We now define our transformation F, on 'p by relativizing the quantifiers and free variables (as if they were universally quantified) to the elements of A defined by the tagging:

TAX)=

W l , t2>[

t , Zt,& S ( x , t l ) &

S(X,t2)

& ~ y ( S ( y , t l ) v S ( y , t , ) ~= y

Figure 1.1

41.

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Theories of reducibility orderings

Figure 1.2

We then replace R ( x , y ) in the matrix by ( R ( x , ~ ) ) ~ I :

=3u,u, w [ S(x, u ) & S(x, u) & S(u,u)

(R(x,y))F’

& S(u, w)& S( w,y) & o P w ] .

It is clear that 8 ’ ~ ( R ( ~ , y ) ) ~ ’ ~ ~ for ~ Rx,y ( x€ , Ay and ) so our transformation F , is as required. (Note that we use two tags rather than one to distinguish the elements of A not in the field of R as well. The point w is included to allow for either R ( x , x ) or l R ( x , x ) for each x E A . ) The next step in our translation project is to show how to code an n-ary relation P as a binary one. The pictures for n = 3 and n = 4 are given in Fig. 1.2. Here the arrows u+y, indicate that R(u,y,) holds. Given then a structure 8 = (A, P ) with P n-ary we construct A’ by adding on to A 1 Z,: ,i new elements for each n-tuple x’ such that P(Z) holds and defining R as indicated by the arrows above. Again, for those with a love of indices, for each x’ such that P ( 3 we add new elements u ( Z ) andyi(Z) for each1 < j < i, 1 < i . We now transform a sentence ‘p about 8 into one c p F 2 about 8’ by relativizing all quantifiers and free variables to the subset A of A’ which is defined by TR(x)=Vy 1 R ( x , y ) and then replacing P ( x , , , . ., x n ) by ( P (x 1’ ...,xn))F2:

+

( P( xl, .. .,x,))“’

= 3u 3 y ;3 y ,,y;, 3 y ; . . .

Again it should be clear that for xi E A %kP(x,, . . . x n ) iff 8 ‘ k ( P ( x l ,...,x,JF2) and so 8k’p&X’krpF2. Indeed, although we do not need it here, an only slightly more complicated procedure (attaching an i-cycle to u to indicate the ith predicate) gives a transformation coding an

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arbitrary string Pl,P2,...Pi...of ni-ary predicates as a single binary one. This procedure appears in RABINand SCOTT(n.d.) . Let us turn now to second order logic by which we mean the usual first order logical language augmented by variables P,,,ranging over all n-ary predicates. We call the language of second order logic C2. Note first that our transformations F, and F, could equally well be applied to structures with many predicates translating P I ,P,,...,P,,,into R,, .,.,R,,, and then into S,,...,S,,, respectively except that we must arrange for the domains to coincide. We can therefore slightly modify a combined transformation to handle our second order language. Without loss of generality we may start with a formula 'p in a prenex form in which all the second order variables are at the beginning of the formula. We wish to specify one subset as the intended domain for all our coded predicates and another as our stock of "new" elements for filling in the required configurations. To simplify matters we will only consider infinite structures. We build all of these ideas into one sentence:

Ic/ v y 3!x [ x #y & R( x , x) & R( y ,x ) ] & Vx 3 ! y [ y #x & R( y ,x) ] & V y 3 ! [~X # Y &

-I

R(x,x)&R(y,~)].

We will think of R ' = { x l R ( x , x ) } as our domain and R" = { x l R~( x , x ) } as our stock of coding elements. We transform a formula 'p of C2 into one cp 4' with only binary relation symbol by setting cp F 4 = 3R ($ &cp '3). ' p F 3 is gotten by first replacing all second order quantifiers 3P,,i(VPn,i)by 3R,,i (VR,,i) which range over binary relations only. Then all first order quantifiers and free variables are relativized to R' and all atomic formulas P,,i(Z)are replaced by (Pn,i(Z))Fzwith the new quantifiers restricted to R " . If one believes in the first order translation given by F, it is not hard to see that 'p is satisfiable (valid) in some infinite structure A iff cpF4 is also. (Note that a second order formula is satisfiable in one infinite structure if it is satisfiable in any structure of the same cardinality. Thus we may keep the set A fixed in the second order case.) Finally we reduce a formula 'p of C2 to one ' p F with only s.i.b. relations. 'pF is just 3S($Fl&'pF5), ' p F 5 is gotten from ' p F 3 (which we think of as . is ( 3 x ) ( R ( x , x ) & . . )) as one might being fully written out, i.e. (3~)~'expect. We replace the second order quantifiers 3R,,i (VR,,i) by 3Sn,i(VSn,i).First order quantifiers are relativized to the elements appropriately tagged by S. Atomic formulas are replaced by their images under F,. Again the correctness of the translation in the first order case easily convinces one that for any infinite A and any 'p of ,?I Ak'p iff AkcpF.

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2. s.i.b. relations and distributive lattices In this section we will show how to pick out in a first order way certain distributive lattices for which quantification over ideals can be used to replace quantification over all s.i.b. relations on the minimal elements (atoms) of the lattice. Combining this with the results of Section 1 will give us a transformation G such that for any sentence 'p of C2 there is an infinite Ak'p if and only if there is a distributive lattice C (always assumed to have a least and a greatest element) with Ck'p'. ('p' will be in the language LZ, i.e. lattice theory augmented by second order monadic quantification over ideals In,,.)This is really just an example of a more general phenomenon of restricted versions of monadic quantification giving full second order logic. Another example that we will discuss briefly below is the theory of commutative rings with quantification over ideals. These results for rings and lattices contrast quite sharply with those of Rabin for Boolean algebras. RABIN(1969) shows that the theory of countable Boolean algebras with quantification over ideals is in fact decidable. Let us consider some (distributive) lattice C with atoms {a,}. The key property we need is independence:

Definition 2.1. A subset X of finite F G X , x < V F i f f x E F .

e is independent if

for any x E X and any

The point is that if some set X such as the atoms of C is independent then quantification over ideals of C gives monadic quantification over X. One just translates a subset of X as the ideal it generates. To move up to quantification over s.i.b. relations over A , the set of atoms, we want our lattice to have some sort of code c(a,,aJ for each unordered pair of distinct atoms { a , , ~ , } .If these codes form an independent set, then quantification over ideals gives monadic quantification over the set of codes which in turn is obviously equivalent to quantification over s.i.b. relations on A : S ( a j , + ) - c ( a i , ~ ) E I swhere I , is the ideal generated ) } , course we will want the codes to be effectively by { c ( a , , + ) ~ S ( a j , a j Of definable to give our final translation. To this end we will specify that (2.1)

for each pair of distinct atoms a, and a, there will be a unique element of the lattice which is strictly above a,Va, but not above any elements other than a,,a,, a,Va2, and zero.

The next step is to guarantee that A and C , the set of codes, are independent subsets of L. We can do so directly in LI (see 4.3)or we can

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even do it in a first order way by requiring that L contain “pseudo-complements” Z,F (not necessarily unique) for each a E A and c E C . By this we mean the following: (2.2) and

(Val E A)(3Z,)(Va2E A ) ( a ,
(VC, E C)(3C1)(VC2E C)(C, < C , ~ C , # C , ) . (2.3) As both A and C are definable these are sentences of lattice theory.

( C ( x ) = ( 3 a , , a ,E A ) [ a , #a,&a,Va, < x & ( V y [y
> 0)

ory=a, ory=a,va,]).

As there is a unique such x for each {a,,a,} by (2.1) this also shows that the function c(al,a2)is also definable as well. Thus if x E A ( C ) and F is a finite subset of A ( C ) with x F, then x ;f F as required for independence. We now begin to describe our translation of second order logic into LI. The first step is to define a transformation G on (prenex) formulas cp of second order logic. One forms c p F and then replaces every quantifier 3S,,,j(VS,,i) by 3Zfl,;Pifl,;) and each occurrence of S,,,(x,y) by c ( x , y ) E I,,i. The first order quantifiers in c p F are of course all relativized to A as well. Now for cpG to faithfully represent cp it must be interpreted in a lattice satisfying (2.1)-(2.3). Note that any c p F is satisfiable only in infinite domains and so if cb(2.1)-(2.3)&cpGG, I must have infinitely many atoms. Finally we define cp to be (2.1) & (2.2) & (2.3) & 9,‘. It should be clear that a sentence cp is satisfiable over some [every] infinite (cardinality K > W ) domain iff c p H is satisfiable in some [every] distributive lattice satisfying (2.1)-(2.3) (with K many atoms). Or at least this should be clear assuming such distributive lattices exist. To answer even such concerns we give a concrete (effective) field of sets representation of the standard countable lattice we have in mind: Let N denote the natural numbers and let C be any (recursive) one-one onto map from [ El2, the pairs of even natural numbers, to 0,the odd natural numbers. Our lattice Lo&2N is the one generated by u and n from the elements {XI, { x , ~ , C ( x , y ) }E, - { x } and N - { C ( x , y ) } . We have thus proven the following:

v

Theorem 2.2. The theory of distributive lattices in the language LI is recursively isomorphic to that of full second order logic (on infinite structures). The restriction to infinite structures can of course be eliminated by slightly complicating the translation to restrict quantification to a definable

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subset of the atoms but this is not too important. We also immediately have similar results for many other restricted monadic theories in which distributive lattices can be interpreted. A rather trivial example is that of the full monadic theory of a binary relation. Actually this particular result follows quite easily from the first order codings alone and was surely known to many of the workers in this area. We mention it explicitly for the sake of our propagandistic intentions and a specific example: THOMASON (1975) showed that the monadic theory of one binary relation can be interpreted in the usual tense and model logics. He remarks that this is a substantial portion of second order logic. These observations show that it is in fact all of second order logic. We should also note that S. KRIPKE (private communication) proved in 1976 that various fragments of a number of modal systems ( S 4 , M , B , K , K 4 , etc.) with propositional quantifiers are recursively isomorphic to full second order logic using some related coding schemes.

3. Reducibility orderings Our next goal is to give a faithful translation of LZ into the first order theories of certain types of partial orderings. Let 9 be a partial ordering. By a translation J of LZ into the first order theory of 9 we mean an effective map from formulas 'k of LZ to ones @ of the language of partial orderings. We call J faithful (with respect to countable structures) if for all 'k, 'k is satisfiable in some (countable) distributive lattice iff @ is satisfiable in 9.We will specify conditions on 9' which guarantee the existence of such a faithful translation but first some definitions for upper semi-lattices. Definition 3.1. An ideal I in

9 is a non-empty subset of 9 such that

x < y E Z implies that x E Z and x,y E Z implies that x v y E I . Every pair ( x , y ) from 9 determines an ideal ZXa = { z E 9 Iz < x & z < y } the pair ( x , y ) is called an exact pair for Z if Z = Z x y . The segment [u,u] of 9 is the set {zE9~u
Now for our conditions on

(3.1) (34 (3.3)

9.

9 is an upper semi-lattice with least element 0. Every countable ideal in 9 has an exact pair in 9. Every countable distributive lattice C is isomorphic to some segment of 9.

Although we could get by with weaker conditions, (3.1)-(3.3) will simplify the exposition. For the translation from LZ,condition (3.3) tells us

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that every countable distributive lattice C can be realized as a segment of 9.Condition (3.2) then says that we can replace (second-order) quantification over ideals of C by (first order) quantification over pairs from 9. More formally, given \k, a formula of LZ, we transform it into \kJ= 3 u , u ( [ u , u ] is a distributive lattice and W) where \k' is obtained as follows: First replace V and A by their usual definitions in terms of <. Then relativize all first order quantifiers to [u,u]. Finally replace all second order quantifiers (31) by ( 3 x ) ( 3 y ) and atomic formulas a E 1 by a < x & a G y . It is clear from conditions (3.1)-(3.3) that a sentence x is true in some countable distributive lattice iff \kJ is true in 9. Our final transformation of second order logic to the first order theory of some partial orderings 9 is thus given by ~ ~ + r p ~ = ( r p ~Stringing )~. together our stated equivalences now shows that a sentence rp of second logic is true in some countable set iff rp is true in 9.It will be useful for some of the applications discussed in Section 4 to observe that we only need the embeddability of one lattice satisfying conditions (2.1)-(2.3) as a segment of 9. We have thus proven the following:

Theorem 3.2. Zf 9 satisfies conditions (3.1), (3.2) and (3.3) (at least for some distributive lattice satisfying (2.1)-(2.3)), then satisfiability (validity) in countable structures for second order logic is one-one reducible to the first order theory of 9. Remark 3.3 It is well-known that the theory of true full second order arithmetic (Th2((N , = , + , X ,)) is reducible to second order logic on countable structures. One simply uses the usual second order definitions of an a-ordering and the standard inductive definitions of and x from the ordering. Indeed if one codes n-tuples as numbers either is equivalent to the monadic second order theory of true arithmetic (i.e. T h ( ( N , 2N,E, < ,+ , X ))) usually referred to as (true) second order arithmetic or (true) analysis. So another rephrasing of Theorem 3.2 is that the truth set of analysis is one-one reducible to Th((P, < )) for any partial ordering of the specified type.

+

Now we will apply known theorems on the structure of various reducibility orderings (with some minor variations) to characterize the exact degree of their theories as that of second order arithmetic.

Theorem 3.4 The second order theory of true arithmetic is recursively isomorphic to the first order theories of the degrees of the following reducibilities: one-one, many-one, truth-table, weak-truth-table, Turing and arithmetic.

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Proof. In one direction it is obvious that all of these orderings are effectively definable in second order arithmetic and so their theories are one-one reducible to that of true analysis. For the other direction we need only show, except for the one-one degrees, that the hypothesis of Theorem 3.2 are satisfied. (The one-one degrees present special problems which we will deal with later.) The other orderings are trivially upper-semi-lattices and so it suffices to verify conditions (3.2) and (3.3). The required embedding theorems are given by LACHLAN (1970) for many-one degrees, LACHLAN(1968) for Turing degrees and HARDING(1974) for the arithmetic degrees. The embedding results for truth-table and weak-truth-table degrees (introduced in Friedberg and ROGERS(1959)) are also essentially given by most proofs of the embedding theorem for Turing degrees with perhaps some minor modifications. We assume familiarity with some exposition of this theorem (as in EPSTEIN (1979) or LERMAN (1981)) and sketch a proof of the following: Lemma 3.5. Every countable distributive lattice can be embedded as an initial segment of the tt and wtt degrees.

Proof. (sketch): The key point is to use trees T, such that for each n if

vz(")(n)is defined for any u, then qF(T)(n)is defined for all 7 of length n. It

is then easy to see that if A is the representative of the top degree constructed and gf. is a characteristic function, then (p," = , A ( X ) for some recursive X C N representing an element x of the lattice (n E A ( X ) iff the nth element of X is in A.) The usual proof shows that q,,"= T A ( X ) but the reduction procedures can be made total quite simply. In the recovery of A ( X ) from (p," if we hit a branching on the appropriate tree T, which differs on X, then it must split for somey. If one computation agrees with Q I ~follow it. If neither, then output zero for all further calculations. To compute cp,"(n) from A ( X ) go to a level of T, of length n and compute cpeqT)for any 7 agreeing with A ( X ) up to this point if there is one to get the required answer. If all such T ( T )are incompatible with A ( X ) output zero from now on. These procedures are clearly total and applied to (p," give the same results as the usual ones. For more details we refer the matter to EPSTEIN (1979) or LERMAN (1981). Thus Turing reducibility and truth-table reducibility coincide (and so also with weak truth table reducibility which is caught between them) on the sets T-reducible to A. The initial segments of these degrees below A of course are also then the same and we have our embedding theorem for tt and wtt degrees. Turning now to condition (3.2) we note that for Turing degrees this is just the theorem of SPECTOR (1956). Essentially the same proof, however,

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gives the result for many-one, truth table and weak truth table degrees. (We include the proof so as to be able to refer to it for the case of one-one degrees.) Let (pe be a list of the appropriate reduction procedures and let I = {u,} with representative sets Ai.We construct X and Y in stages. At stage s = (sI,s2) we have X , and Y, defined on the first s - 1 columns of X and Y and on at most finitely many other places. We ask if some finite extensions X’ of X , or Y ’ of Y, guarantee that cp:’(x)#cpT(x) for some x. If so choose such extension. If not let X’=X,, Y ’ = Y,. We now get X , , , and Y,,, by extending X’ and Y ’ so that their sth column (the elements such that ( s , y ) belongs to them) is equal to A , except perhaps at the finitely many points of the column which are already fixed. The point of course is :cp r, then at stage s=(sl,s,) any extension of X , or Y, would that if cp= give the same set and so if we fill out X , to Z by letting Z ( n ) = 0 when X , is not defined at n, then we have cp”, but Z is even many-one reducible to { A , l i < s } so that the degree of cpc‘is in I . We now have to verify (3.2) for the arithmetic degrees. We assume familiarity with perfect forcing as in SACKS(197 1): Let { A , } be the sets with arithmetic degree in the given ideal I . Let G , x G, be generic with respect to the product of the partial ordering of pointed perfect closed sets arithmetic in some A; with itself. As the conditions are pointed, A, is arithmetric in GI and G, for each i. Standard arguments now show that if T is arithmetic in both GI and G,, then T is arithmetic in some A , : If so some condition

(P,,P,)I~Vx(cp~~”(x)= c p2 y (4

for some P , , P , and n. By the definition of forcing TI”)

‘pe,

(x)= c p w :

for any T IE P I ,T2E P2. Thus T is arithmetic in any such T I or T, but we can choose T I ,T, arithmetic in P I ,P, and so in some A . Thus the degree of T is in I. This completes our proof except for the one-one degrees whch are a bit more complicated. The first problem is that by YOUNG(1964) they do not form an upper semi-lattice. We could have proven our main theorem for directed sets but this is not the real source of difficulties. The main problem arises in the verification of (3.2). Following the outline above we build sets X and Y for a given sequence of sets { A i } using 1-1 reduction procedures. We then get that if Z is one-one reducible to X and Y , then it is reducible to the disjoint union of Bifor i
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Now if O
-

C/Cj =

c

c;/c,

i
as cj is indecomposable. Of course c,/cj = 0 for i # j and so c / c j = 1 or 00 if cj is a cylinder. But clearly V c , / c j > 1 and is 00 if cj is a cylinder. Thus c < v c , . As the initial segment below V c i is a lattice and c, < c for each i we have the desired conclusion that c = @ c, = v c , . Theorem 3.2 can also be used to give a lower bound on the complexity of degree orderings which are not necessarily definable in second order arithmetic.

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Corollary 3.6. True-second order arithmetic is one-one reducible to the first order theory of a-degrees for every countable admissible a.

Proof. By MACINTYRE (1974) (3.3) holds for each countable admissible. The standard proof of Spector's theorem using a-reduction procedures for the 'p, and a-finite extensions as well then gives (3.2) for a-degrees. It is tempting to try to apply these arguments to hyperdegrees and constructibility degrees as well. With mild set theoretic assumptions (e.g. is countable for every real x ) (3.1) holds for the constructibility (1977). For hyperdegrees, finite distributive lattices degrees by ADAMOWICZ (1970), but it seems likely that the have been embedded by THOMASON countable ones can also be embedded. The real problem is that (3.2) fails for these degrees. We believe that it is probably possible to find an embedding in the constructibility degrees of a lattice of the required sort with enough genericity to do something like the proof for the arithmetic case. Although this would suffice for some of our later applications it is not enough to get the main theorem. The problem is that we do not see how to guarantee the required genericity by any first order statement about the lattice that we embed. 4. Applications

We will give a couple examples of applications of the methods of Sections 1 and 2 to other subjects and a sketch of some further results that can be gotten for the Turing degrees. Our first example is the theory of commutative rings in the language RZ which augments the usual first order language of ring theory with quantification over ideals. In analogy to the codings of Section 2 we show how to code s.i.b. relations by ideals to give a reduction of second order logic to this theory. The role of the top element of our lattice is now played by an element a of a ring and the atoms are replaced by the set D = { x l x 2 = a} which will be the domain for our s.i.b. relations. Corresponding to condition (2.1)-(2.3) we have the following: (4.1)

For all distinct elements xl, x2,x3,x4 of D, x,x,#x,x,.

(44

(Vx

17

x2,

x3)D( xIx2x3 = 0).

(4.3) (Vx, # ~ 2 ) D ( ~ z ) ( V x ~ # x 4 ) D ( x 3 x ~ € I & xI). ,x2~ Quantification over s i b . relations on D is replaced by quantification over ideals in the ring by the coding S(x,, x,)wxix, E 1. Condition (4.3) guarantees that for each relation S there is an ideal I with S(xi,xj)exixj

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195

€ I . Thus we can quantify over all s.i.b. relations on D in RZ and so encode all of second order logic. (Again to see that rings satisfying (4.1)-(4.3) exist just consider the formal linear combinations over Z of 1,a, xi, +xk with the indicated multiplication rules i.e. the “semi-group” ring over Z.) We next consider the theory of the isols. ELLENTUCK (1970) first noticed that one could express certain second order properties of A (the Q,-theory) in the first order language of (A, +, X , a x ) . He then (1973) eliminated exponentiation and also showed by a coding of arithmetic that the first order theory of (A, +, X ) is recursively isomorphic to second order arithmetic. NERODEand MANASTER (1971) later announced that this can also be proved for the theory of (A, < ) by using coding t e c h q u e s from NERODE and MANASTER (1970). In fact it also. follows immediately from the results of that paper and those of Section 1 much the way those of Section 3 do. Again the key ideas are ideals and covers for them although (1970) is not so algebraic. We the terminology of NERODE and MANASTER supply a glossary. Definition 4.1. Z L A is a normal ideal iff (1) l,OEI, (2) x < y E I - x E Z , and (3) x,y E Z-x + y E I . Definition 4.2. X is an indecomposable cover for I iff (1) WY E O ( x >Y), and (2) x = y + z + y ~ Z o r z ~ Z . NERODE and MANASTER (1970, $1) shows that every normal ideal has 2’0 pairwise incomparable indecomposable covers (which are minimal elements modulo I ) . They then show how to describe an is01 T (analogous to the top of our distributive lattice initial segment of 9)with countably many minimal elements over the definable ideal w below it as well as other covers coding pairs of these minimal elements. These “minimal” elements or indecomposable covers serve as the domain on which they code binary relations by covers of other ideals generated by sets of code elements. (Actually as they code arbitrary binary relations rather than s.i.b. ones they are forced up one level of complexity more than is necessary.) As these covers are elements of A and all the required predicates on A are definable from the ordering they actually have coded quantification over binary relations on a countable domain in the first order theory of (A, <). It is therefore recursively isomorphic to second order arithmetic.

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To close this paper we would like to sketch some applications to the and SHORE (1980). Turing degrees that will be treated in full in NERODE Using the first order codings of Section 1 we can extend the description of our lattice in Section 2 to require that certain elements of the lattice code (via the principal ideals they generate) an d i k e ordering on some subset of the atoms together with operators satisfying the inductive definitions of and x. By quantifying over all of q and so over all subsets of the ordering we can also require that it be a true w-ordering. If d is the top element of such a lattice we say that d or the set D of degree d codes a model of true arithmetic. A straightforward calculation shows that there is a function f < D ( 5 )with d, =deg{ f ( n ) } D the degrees interpreted as n in this model of arithmetic. The key observ'ation needed for our applications is the following:

+

Lemma 43. (a) Zf the pair a,,% codes the set A in this model, i.e. n E A ( = ) d , < ~ a , , a , , then A O". We give some sample applications of this observation. First a definability result. Let &(x)=x is arithmetic. By Lemma 4.3 (a) if any set A is coded in a model below 0" by a pair recursive in xVO", then A is arithmetic in x. By Lemma 4.3 (b), if x is not arithmetic, then there is a model below 0" and a pair recursive in xV0" coding a non-arithmetic set on this model. Thus we have a first order definition of & from the parameter 0". Similar arguments prove the following:

c

Theorem 4.4. If C 9 is closed downward and under jump and join, then C is first order definable in (9, < , O M > iff C is definable in second order arithmetic. A bit more work would enable us to give relativized versions of the above and replace the parameter 0" by the predicate &. Somewhat different methods can be employed to prove the following strengthening of the main result (Theorem 3.12) of SIMPSON (1977): Theorem 4.5. (a) The a-jump is definable in (9, < ,O"). (b) If C {xlx > O'"'}, then C is definable in second order arithmetic is definable in (9, < ,0" ).

iff

c

197

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We now consider theories of the Turing degrees with the jump operator. Again by analyzing which sets can be coded in models of arithmetic below given degrees as in Lemma 4.3 we can show that for most degrees a,qa = ({ XIX > a}, < ,’) is not elementarily equivalent to 9‘ = (9, < ,’). A more complicated version of Spector’s theorem gives quite a good estimate of what can be coded and lets us prove the following improvement of (1977) and EPSTEIN (1978). another result from SIMPSON

Theorem 4.6 If qa’ = 9’, then a(4)< O(’). We can also use these computations to show that (9, < ,( n ) ) S (9, n#m. ((n) is the nth jump operator.) The idea is that if n > m , then enough iterations gives us a large enough gap between O(kn) and O(&“‘) so that certain definable sets can be coded by pairs in models of arithmetic below O(kn) but not below O(km). This answers a question of SELMAN(1972). For our last application we consider automorphisms of 9. Now it is not known if there are any non-trivial automorphisms of 9‘ or even of 9. For q’it is known that every automorphism is the identity on the cone above 0(3)(RICHTER(1977) after JOCKUSCH and SOLOVAY(1977) and YATES (1971)). By combining Lemma 4.3 with the completeness theorem of FRIEDBERG (1957) we can prove the following:

< , ( m ) ) if

Theorem 4.7. Every automorphism cp of 9 is the identity on a cone.

Proof. If x>cp-’(O”)=c, then Lemma 4.3 tells us that cp(x) c”). So by Friedberg x = y ( ’ ) = ~ V c ( ~ )for some y > c. Thus cp(x) = (P(Y(’)) = cp(Y)Vcp(C(’))

Y(’)VQJ(C(’)).

-’

Thus if x > cp(c5) as well, cp(x) < x. Applying the same argument to cp we see that cp-’(x)<(x (and so x
Theorem 4.8. Every automorphism of above O(3).

9fixing

0’ is the identity on the cone

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As our departing salvo we remark that although on its face picking out a copy of true w depends on quantification over all subsets and so over all of 63 this can in certain cases be avoided by calculating just how much we need to assert to guarantee well-foundedness. This allows us in NERODE and SHORE(1980) to prove analogs for all our results for substructures of the degrees closed downward and under jump and join.

References ADAMOWICZ, z. [ 19771 Constructible semi-lattices of degrees of constructibility, in: Set Theory and Hierarchy Theory V , Lecture Notes in Mathematics, 619, edited by Lachlan, Srebny and Zarach (Springer-Verlag, Berlin), pp. 1-44. CHERLIN, G. [ 19761 Model Theoretic Algebra, Selected Topics, Lecture Notes in Mathematics, 521 (Springer-Verlag, Berlin). CHURCH,A. and W. V. QUINE [ 19521 Some theorems on definability and decidability, J. Symbolic Logic, 17, 179-187. ELLENTUCK, E. (19701 A coding theorem for isols, J. Symbolic Logic, 35, 378-382. [I9731 Degrees of isolic theories, Notre Dame J. Formal Logic, 14, 33 1-340. EPSTEIN, R. [1978] Analysis and the degrees of unsolvability G O , Notices Am. Math. SOC.,25, A-441. [1979] Degrees of Unsolvability: Structure and Theory, Lecture Notes in Mathmetics, 759 (Springer-Verlag, Berlin). A. D. TAIMANOV and A. M. TAITSLIN ERSHOV,Yu., I. A. LAVROV, [ 19651 Elementary theories, Russian Math Surutys, 20, 35- 105. FREGE,G. [ 18791 Begriffschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle. Translated in: From Frege to Godel, edited by J. van Heijenoort (Harvard Univ. Press, Cambridge, MA, 1967), pp. 1-82. R. M. FRIEDBERG, [1957] A criterion for completeness of degrees of unsolvability. J. Symbolic Logic, 22, 159-160. FRIEDBERG, R. M. and H. ROGERS [ 19591 Reducibility and completeness for sets of integers, 2. Math. Logik Grundlagen Math., 5, 117-125.

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HARDING,C. J. [ 19741 Forcing in recursion theory, Thesis, University College of Swansea. HIRSCHFELD, J. and W. H. WHEELER [ 19751 Forcing, Arithmetic, Division Rings, Lecture Notes in Mathematics, 454 (Springer-Verlag, Berlin). JOCKUSCH, C. G . and D. POSNER [1978] Double jumps of minimal degrees, J. Symbolic Logic, 43, 715-724. JOCKUSCH, C. G. and R. M. SOLOVAY [1977] Fixed points of jump preserving automorphisms of degrees. Israel J . Math., 26, 91-94. KLEENE, S. C. [ 19551 Hierarchies of number theoretic predicates, Bull. A m . Math. SOC., 61, 195-213. KLEENE,S. C. and E. L. POST [ 19541 The upper semi-lattice of degrees of recursive unsolvability, Ann. Math., 59, 379-407.

LACHLAN,A. H. [1968] Distributive initial segments of the degrees of unsolvability, 2. Math. Logik Grundlagen Math., 14, 457-472. [1969] Initial segments of one-one degrees, Pacific J. Math., 29, 35 1-366. [1970] Initial segments of many-one degrees, Can. J. Math., 22, 75-85. LAVROV,I. A. [ 19631 Effective inseparability of the sets of identically true formulae and finitely refutable formulae for certain elementary theories, Algebra i Logika, 2, 5-18. LERMAN,M. [1978] Initial segments of the degrees below 0', Notices A m . Math. SOC., 25, A-506. [ 19811 The Degrees of Unsoluabifify (Springer-Verlag, Berlin), to appear. MACINTYRE, J. M. [ 19731 Non-initial segments of a-degrees, J. Symbolic Logic, 38, 368-388. NERODE, A. and A. B. MANASTER [ 19701 A universal embedding property of the RET's. J. Symbolic Logic, 35, 51-59. [1971] The degree of the theory of addition of isols, Notices A m . Math. SOC.,18, 83.

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NERODE,A. and R. A. SHORE [ 19801 Reducibility orderings: theories, definability and automorphisms, Ann. Math. Logic, 18, 61-89. M I N , M. [1965] A simple method for undecidability proofs and some applications, in Logic, Methodology, and Philosophy of Science, Proc. 1964 Int. Congress. edited by Y. Bar-Hillel (North-Holland, Amsterdam), pp. 58-68. [1969] Decidability of second order theories and automata on infinite trees, Trans. Am. Math. Soc., 141, 1-35. [ 19771 Decidable theories, in: Handbook of Mathematical Logic, edited by J. Barwise (North-Holland, Amsterdam), pp. 595-629. RABIN,M. and D. SCOTT [n.d.] The undecidability of some simple theories, mimeographed notes. RICHTER,L. J. C . [1977] Degrees of unsolvability of models, Ph. D. Thesis, University of Illinois, Urbana-Champaign. SACKS,G. E. I19711 Forcing with perfect closed sets, in: Axiomatic Set Theory, Proc. Symp. Pure Math. vol XVII pt I , edited by D . Scott (Am. Math. SOC.,Providence, RI). SELMAN, A. L. I19721 Applications of forcing to the degree theory of the arithmetic hierarchy, Proc. London Math. SOC.,25, 586-602. SIMPSON, S. G . [1977] First order theory of the degrees of recursive unsolvability, Ann. Math, 105, 121-139. SPECTOR,C . [1956] On degrees of recursive unsolvability, Ann. Math., 64, 581-592. THOMASON, S. K. [1970] On initial segments of hyperdegrees, J . Symbolic Logic, 31, 159- 168. [ 19751 Reduction of a second order logic to modal logic, Z . Math. Logik Grundlagen Math., 21, 107-1 14. YATES,C. E. M. I19711 Initial segments and implications for the structure of degrees, in: Conference in Mathematical Logic-London 1970, edited by W . Hodges, Lecture Notes in Mathematics, 255 (Springer-Verlag, Berlin), pp. 305-335. YOUNG, P. R. [ 19641 On reducibility by recursive functions, Proc. Am. Math. SOC.,15, 889-892.