Interpreting N in the computably enumerable weak truth table degrees

Annals of Pure and Applied Logic 107 (2001) 35–48www.elsevier.com/locate/apal

Interpreting N in the computably enumerableweak truth table degrees

Andr�e Nies1Department of Mathematics, The University of Chicago, 5734 S. University Av., Chicago,

IL 60637, USA

Received 10 November 1999; received in revised form 27 January 2000; accepted 5 February 2000

Communicated by R.I. Soare

Abstract

We give a �rst-order coding without parameters of a copy of (N;+;×) in the computablyenumerable weak truth table degrees. As a tool, we develop a theory of parameter de�nablesubsets. c© 2001 Elsevier Science B.V. All rights reserved.

MSC: primary 03D25

Keywords: Weak truth table degrees; True arithmetic; Uniform de�nability

0. Introduction

Given a degree structure from computability theory, once the undecidability of itstheory is known, an important further problem is the question of the actual complex-ity of the theory. If the structure is arithmetical, then its theory can be interpretedin true arithmetic, i.e. Th(N;+;×). Thus an upper bound is ∅(!), the complexity ofTh(N;+;×). Here an interpretation of theories is a many–one reduction based on acomputable map de�ned on sentences in some natural way. An example of an arith-metical structure is DT (6 ∅′), the Turing-degrees of �02-sets. Shore [16] proved thattrue arithmetic can be interpreted in Th(DT (6 ∅′)). A stronger result is interpretabilitywithout parameters of a copy of (N;+;×) in the structure (interpretability of structuresis de�ned in [8, Chapter 5]). The main purpose of this paper is to prove such a resultfor the structure Rwtt of computably enumerable weak truth table degrees. So far theundecidability of Th(Rwtt) is known [3]. This result brings a program closer to itscompletion which has been carried out by various researchers over the past years: to

1 Partially supported by NSF grant DMS-9803482.E-mail address: [email protected] (A. Nies).

0168-0072/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.PII: S0168 -0072(00)00021 -X

36 A. Nies / Annals of Pure and Applied Logic 107 (2001) 35–48

determine the complexity of the theory for structures from computability theory. Wediscuss some results. For the c.e. many–one and Turing degrees, it has been provedthat a copy of (N;+;×) can be interpreted without parameters ([12] and [15], respec-tively). For the c.e. truth-table degrees and the lattice E of c.e. sets under inclusion,interpretations of Th(N;+;×) in the theory have been given (for the �rst, see [14];the second result is due to Harrington, see [7]). In E one cannot interpret a copyof (N;+;×) [7], which shows that the stronger, model-theoretic result is not alwaysimplied by the mere interpretability of the theory of (N;+;×). For the structures Rm

and RT of c.e. many–one and c.e. Turing degrees as well as for E, the methods em-ployed (usually auxiliary codings of copies of (N;+;×) with parameters and uniformde�nability results) have been used to obtain further results of a model-theoretic avorabout the structure (see the same references). For RT one obtained ∅-de�nability ofall jumps classes except Low1. For both Rm and RT, restrictions on automorphismswere derived: each automorphism of Rm is arithmetical on any proper initial interval,and, dually each automorphism of RT is arithmetical on any proper end interval (see[13] for the latter result). Finally, for E one obtained elementary nonequivalence ofrelativizations. We hope that the new coding methods eventually lead to such resultsfor Rwtt.Degree structures where so far only undecidability of the theory is known include the

c.e. Q- and btt-degrees [6] and [11], as well as the enumeration degrees of �02-sets [17].Among the degree structures induced on c.e. sets, only Rwtt and Rm are distributive

as an upper semilattice, namely they satisfy

∀x∀a∀b[x6a ∨ b ⇒ ∃a06a∃b06b x = a0 ∨ b0]: (1)

See Lachlan [9] for a proof in the nontrivial case of Rwtt. As a tool for proving ourmain result, we develop a theory of two sorts of parameter de�nable subsets, usingthe distributivity of Rwtt in an essential way. One of them is the class of EN-sets.EN-sets are relatively de�nable without parameters in an end segment, i.e. an upwardclosed subset E of Rwtt, while E is de�nable from two parameters c; d . The numbern∈N will be represented by (parameters de�ning) any EN-set of size n, but theremay also be in�nite EN-sets. Using the combinatorics of EN-sets, we give �rst-orderde�nitions in terms of parameters for whether two EN-sets have the same size, and ofternary relations corresponding to the operations + and × which behave properly on the�nite EN-sets. For instance, for +, we express that an EN-set is the disjoint union oftwo others.The second type of uniformly de�nable set, called ID-set (“ID” stands for ideal) is

needed to single out the �nite EN-sets in a �rst-order way. The concept of ID-setsis dual to that of EN-sets. Thus ID-sets are relatively de�nable without parameters inan initial segment, actually in an ideal I , while I is de�nable from two parametersc; d . To single out �nite EN-sets, we will compare EN-sets to ID-sets, using uniformlyde�nable 1-1 maps between the �rst and the second.The idea of representing a number n in a natural way by the class of EN-sets of that

size sets our proof apart from the ones used for the other structures discussed above,

A. Nies / Annals of Pure and Applied Logic 107 (2001) 35–48 37

which make use of auxiliary codings of copies of (N;+;×) with parameters. However,this idea was �rst used in [11] for the upper semilattice of c.e. equivalence relationsmodulo �nite di�erences.An important fact is that there is an easy way to produce �nite EN-sets: each �nite set

of low degrees which pairwise join up to greatest c.e. wtt-degree is an EN-set (this usesan idea of Ambos-Spies). We �rst use this fact to give a quite elementary new proof ofundecidability for Th(Rwtt). Slightly re�ning the proof yields the undecidability of �5-Th(Rwtt) as a partial order. In Lempp and Nies [10] a coding of �nite bipartite graphsbased on ID-sets is developed, which even yields the undecidability of �4-Th(Rwtt).The �2-theory of Rwtt as a partial order is decidable [2]. Thus, the following problemremains open.

Question 0.1. Is the �3-theory of Rwtt as a p.o., or at least as an upper semilattice,undecidable?

1. Some terminology

We recall some de�nitions. See [15] for more details. For a �rst-order language L,a scheme for coding in an L-structure A is given by a list of L-formulas

’1; : : : ; ’n

with a shared parameter list �p, together with a correctness condition �( �p).

Convention 1.1. If a scheme SX is given, X; X0; X1; : : : , denote objects coded via SX

by a list of parameters satisfying the correctness condition.

Example 1.2. A scheme Sg for de�ning a function g is given by a formula ’1(x; y; �p)de�ning the relation between arguments and values, and a correctness condition �(x; y;�p) which says (at least) that a function is de�ned: ∀x∃61y’1(x; y; �p)

Example 1.3. A scheme for de�ning n-ary relations on A is given by a formula’(x1; : : : ; xn; �p) and a correctness condition �( �p).

De�nition 1.4. A class C of n-ary relations on A is uniformly de�nable in A if, forsome scheme S for coding relations, C is the class of relations coded via S as theparameters range over tuples in A which satisfy the correctness condition.

For instance, if A is a linear order and C is the set of closed intervals, then C is uni-formly de�nable via the scheme consisting of ’1(x; a; b)⇔ a6x6b and the correctnesscondition �(a; b)⇔ a6b.

Notation 1.5. As in Soare [18, p. 49], we assume that the use of the computation{e}As (x), u(A; e; x; s)6s. For e= 〈e0; e1〉 let

[e](x) ' maxy6x

’e1 (y):


Let [e]A(x) be {e0}A(x) if [e](x) and {e0}A(x) are de�ned, and the computation hasuse 6[e](x). Otherwise [e]A(x) is unde�ned. In a similar way, de�ne the approxima-tions at stage s, namely [e]s(x) and [e]As (x).Note that A6wttB⇔A= [e]B for some e. This implies that

{〈e; i〉:We6wttWi} is �03: (2)

2. Uniformly de�nable classes in Rwtt

We develop a coding without parameters of a copy of (N;+;×) in Rwtt. Recall thatwe plan to use two types of uniformly de�nable subsets.We �rst prove some facts which lead to the concepts of EN- and ID-sets. Most of

the facts are algebraic. We expose the duality between the two concepts, as far as thenonsymmetric framework of an upper semilattice which may not be a lattice allowsthis. In the following let (D;6;∨; 0; 1) be a distributive upper semilattice with leastand greatest elements 0; 1.

Lemma 2.1. Suppose that b; y0; : : : ; yn ∈D.1. If b∧yi=0 for each i6n; then b∧ supi yi=02. If b∨yi=1 for each i; then there is t ∈D such that b∨ t=1 and t6yi for each i.

Proof. (i) If 0¡x6b; supi6n yi, then by distributivity, there is an i and r ∈D suchthat 0¡r6x; yi. But then r6b; yi, contrary to b∧yi=0.(ii) If n=0 let t=y0. Else, since y16b∨y0, we can choose a t16y0 and b16b

such that y1 = b1 ∨ t1. Then, 1= b∨ b1 ∨ t1 = b∨ t1, so if n 6=1, y26b∨ t1 implies thatwe can pick t26t1 and b26b such that y2 = b2 ∨ t2 and thus 1= b∨ t2. Continuing inthis way we obtain t= tn6y0; : : : ; yn such that b∨ t=1.

For d0; : : : ; dn ∈D, let

E(d0; : : : ; dn) = {x ∈ D:∀y[∀i6n(y6di)⇒ y6x]}: (3)

Thus E(d0; : : : ; dn) is the set of upper bounds of the ideal [0; d0]∩ · · · ∩ [0; dn]. Notethat di ∈E(d0; : : : ; dn) for each i. Finite EN-sets {p0; : : : ; pn} will be sets which arerelatively de�nable in E(p0; : : : ; pn). First, we need a characterization of the elementsin such an end segment.

Lemma 2.2. For d0; : : : ; dn ∈ (D;6;∨; 0; 1);x ∈ E(d0; : : : ; dn)⇔ x = inf

i6n(x ∨ di):

Proof. For the direction from right to left, clearly x∨di ∈E(d0; : : : ; dn) for each i.Hence, if the in�mum exists, it is also an upper bound for the ideal [0; d0]∩ · · · ∩[0; dn].


For the direction from left to right, the argument is similar to the one used inthe proof of Lemma 2.1(ii). If y6x∨di for each i, then by distributivity we canchoose x06x and q06d0 such that y= x0 ∨ q0. If n¿1, choose x16x; q16d1 suchthat q0 = x1 ∨ q1. Continuing in this way we obtain qn6dn such that qn−1 = xn ∨ qn.Moreover qn6d0; : : : ; dn, so qn6x. Hence qn−16x; : : : ; q06x and �nally y6x.

De�nition 2.3. For x; y∈D, we write nd[x; y] if x¡y and the interval [x; y] does notembed the 4-element boolean algebra preserving least and greatest element.

Clearly nd[x; y] can be expressed in the language of p.o. In the next lemma, (i)leads to the de�nition of EN-sets, and (ii) to the de�nition of ID-sets.

Lemma 2.4. (i) Let p0; : : : ; pn be a �nite sequence of elements of D such that foreach i; nd[pi; 1] (in particular; pi¡1) and for i 6= j; pi ∨pj =1. Then {pi: i6n} isthe set of minimal elements x in E=E(p0; : : : ; pn) such that nd[x; 1].(ii) Let (ai) be a �nite or in�nite sequence of elements of D such that for each i

nd[0; ai] and for i 6= j; ai ∧ aj =0. Then {ai} is the set of maximal elements x in Isuch that nd[0; x]; where I is the ideal of D generated by {ai}.

Proof. (i) It is su�cient to prove that

x ∈ E & nd[x; 1]⇒ ∃j pj6x:

Since x¡1, by Lemma 2.2 there is j such that x∨pj¡1. Moreover, by Lemma 2.1there is t such that, for all i 6= j, t6x∨pi and x∨pj ∨ t=1. We can suppose thatx6t. By Lemma 2.2, x= inf k6n x∨pk , so (x∨pj)∧ t= x. By nd[x; 1], this impliest=1, so x∨pi=1 for i 6= j and x= inf k6n x∨pk = x∨pj.(ii) It is su�cient to prove that

x ∈ I & nd[0; x]⇒ ∃j x6aj:

Since x∈ I , x6 supi6n ai for some n. By distributivity, x= supi6n ai for some ai6ai

(i6n). Since 0¡x, some aj does not equal 0. By Lemma 2.1, aj ∧ supi6n; i 6=j ai=0,so nd[0; x] implies that ai=0 for i6n; i 6= j, hence x= aj6aj.

In the context of Rwtt, we are able to give �rst-order de�nitions with parametersof the set E in (i) of the preceding lemma, and also of I in (ii) if (ai) is a �nite oran in�nite u.c.e. sequence. We use the following theorem of Ambos-Spies, Nies andShore.

Theorem 2.5 (Ambos-Spies et al. [3]). Let I be a �03-ideal of Rwtt. Then there existsa; b∈Rwtt such that I = [0; a]∩ [0; b].

Degrees a; b as above are called an exact pair for I . Note that, conversely, each idealwhich has an exact pair is �03, so that the theorem constitutes a uniform de�nabilityresult for the class of �03-ideals.


Lemma 2.6. (i) Suppose {p0; : : : ; pn} is a subset of Rwtt such that nd[pi ; 1] for eachi and pi ∨ pj = 1 for i 6= j. Then {p0; : : : ; pn} is de�nable from two parameters c; d viaa formula ’P(x; c; d).(ii) Suppose (ai) is a �nite or in�nite u.c.e. sequence in Rwtt such that nd[0; ai]

for each i and ai ∨ aj = 0 for i 6= j. Then {ai} is de�nable from two parameters c; dvia a formula ’Z(x; c; d).

Proof. (i) Observe that I = [0; p0]∩ · · · ∩ [0; pn] is a �03-ideal by (2), so I = [0; c]∩[0; d ] for some c; d . Thus E(p0; : : : ; pn)=E(c; d) is de�nable from c; d via the formula (x; c; d)=∀y[y6c; d⇒y6x]. Let ’P(x; c; d) be the formula expressing that x is aminimal element in {z: (z; c; d)} such that nd[x; 1].(ii) Let I be the ideal generated by {ai}. It follows from (2) that I is �03. So, once

again, I = [0; c]∩ [0; d ] for some c; d . Let ’Z(x; c; d) be the formula expressing that xis a maximal element 6c; d such that nd[0; x].

We are now ready to specify the notions of EN- and ID-sets by appropriate schemesof the same type as in Example 1.3.

De�nition 2.7. (i) Let SP the the scheme given by the formula ’P(z; c; d) and thecorrectness condition �(c; d) expressing that whenever x; y satisfy the formula andx 6=y, then x∨y=1. Subsets of Rwtt coded via SP are called EN-sets.(ii) Let SZ the scheme given by the formula ’Z(z; c; d) and the correctness condition

�(c; d) expressing that whenever x; y satisfy the formula and x 6=y, then x∧y=0.Subsets of Rwtt coded via SZ are called ID-sets.

Notice that subsets of �nite EN-sets are EN-sets themselves. In particular, ∅ is theEN-set coded by c= d = 1. Also ∅ is the ID-set coded by c= d = 0.

3. Undecidability of Th(Rwtt)

We develop a scheme SC to code arbitrary relations between �nite EN-sets. Thecoding methods of this section will also be used to obtain a coding of a copy of(N;+;×).The abundance of EN-sets stems from the fact that each low p∈Rwtt satis�es

nd[ p; 1]. 2 Thus, whenever p0; : : : ; pn are low and pi ∨ pj = 1 for i 6= j, then {p0; : : : ; pn}is an EN-set. For each n, such wtt-degrees p0; : : : ; pn can be easily obtained by themethod of the Sacks splitting theorem (see [18]). In view of later applications, we willprove a more general version of this in Proposition 3.2 below.

Theorem 3.1. If p∈Rwtt is low; then nd[ p; 1].

2 The author would like to thank Klaus Ambos-Spies for suggesting this.


Proof. We slightly modify the proof of an extension of the Lachlan Non-DiamondTheorem in Ambos-Spies [1]. He proves that, if a0; a1; b0; b1 are c.e. Turing degreessuch that a0 ∨ a1 = degT (∅′) and b0 ∨ b1 is low, then, for some i61, ai is not bi-cappable. Here a is called b-cappable if there is a cb such that b= a∧ c. Aninspection of the proof reveals that it can be adapted to wtt-reducibility. (The T -reductions built during the construction have recursively bounded use anyway, and theproof of Lemma 6 [Lemma 9] goes through. In particular, if the reduction proce-dures occurring in requirement Re are now wtt-reductions [e1]B0 and [e2]B1 , then thestep counting functions g in the proofs of those lemmas can be computed from B0[B1] with recursively bounded use. So the weaker hypothesis C0wtt B0 [C1wtt B1]su�ces.)Here we use only the special case of the theorem where b0 = b1 = p. If nd[p; 1] fails,

then there are a0; a1¡1 such that a0 ∨ a1 = 1 and a0 ∧ a1 = p. So for both i=0 andi=1, ai is p-cappable via ci= a1−i.

We now prove the existence of EN-sets relating in a certain way to given degrees.

Proposition 3.2. Suppose that u0; : : : ; um¡1. Then for each n¿0 there exist lowC0; : : : ; Cn ∈Rwtt such that {C0; : : : ; Cn} is an EN-set and ui ∨ Cj¡1 for each i6m; j6n.

Proof. Choose c.e. sets Ui ∈ ui. We construct c.e. sets Vj such that the statement ofthe theorem holds with Cj =degwtt(Vj).To achieve Cj ∨ Cj′ = 1 (j′ 6= j) we ensure that K =Vj ∪Vj′ (where K is some creative

set). For nd[Cj; 1], we make each Vj low and apply Theorem 3.1. We meet the standardlowness requirements

Le; j: ∃∞s{e}Vj (e)[s] is de�ned ⇒ {e}Vj (e) converges:Finally, for ui ∨ Cj¡1 (06i6m; 06j6n) we meet the requirements

Ne; i; j: K 6= [e]Ui⊕Vj ;

by refraining from changing Vj till a permanent disagreement occurs (or [e]Ui ⊕ Vj ispartial). Let (Rk) be some priority listing of the L-type and N -type requirements. If Rk

is Ne; i; j let

length(k; s) = min{x:∀y ¡ x K(y) = [e]Ui⊕Vj (y)[s]

and let r(k; s)= max{[e]s(y): y¡length(k; s).If Rk is a lowness requirement Le; j, the restraint associated with Rk is

r(k; s) = u(Vj; s; e; e; s):

Construction. At stage s+1, if Ks=Ks+1 do nothing. Else, say y is the unique elementin Ks+1−Ks. Determine the minimal k such that y¡r(k; s). If k fails to exist enumeratey into all sets Vj. Else let j be the number such that Rk =Le; j or Rk =Ne; i; j for some e; i.


Then Vj is the set such that enumerating y into Vj would violate r(k; s). So enumeratey into Vj′ , for each j′ 6= j. This completes the description of the construction.Clearly K =Vj ∪Vj′ for j 6= j′. By induction on k we prove:

Lemma 3.3. Let k¿0.(i) The requirement Rk is met.(ii) r(k)= lims r(k; s) exists and is �nite.

Proof. Assume the lemma holds for all h¡k. Choose a stage s0 such that for allh¡k, r(h; s0) has reached the limit, and K does not change below maxh¡k r(h) atany stage s¿s0. Then at no stage s¿s0 can any number y¡r(k; s) enter Vj, wherej is determined from k as in the construction: j is the number such that Rk =Le; j orRk =Ne; i; j for some e; i.If Rk =Le; j, then Rk is met, because if ever {e}Vj [s] converges for s¿s0, then

this computation is preserved. Hence also r(k; s) reaches its limit. Now suppose thatRk =Ne; i; j.For (i), assume for a contradiction that K = [e]Ui⊕Vj . Then lim sup length(k; s) =∞:

We obtain a wtt-reduction of K to Uj as follows: given an input y, compute s¿s0such that length(k; s)¿y and Uid[e](y)=Ui; sd[e](y). Then r(k; t)¿[e](y) for all t¿s,so (by the monotonicity of the function [e]) [e]Ui⊕Vjdy+1 is protected from changingat stages ¿s. So K(y)= [e]Ui⊕Vj (y)[s]. Since ui¡1, we conclude that Ne; i; j is met.For (ii), let x be least such that K(x) 6= [e]Ui⊕Vj (x). Let s1¿s0 be least such that,

[e](x) is de�ned, then K(x) and Ui ⊕ Vjd[e](x) have reached their �nal values at s1.Then length(k; s)6x from s1 on, hence r(k; s) reaches it limit.

It is our next goal to code relations between arbitrary �nite EN-sets.

Proposition 3.4. There is a scheme SC for coding objects of the form (P0; P1; R) inRwtt ; where P0; P1 are EN-sets; which has the following property: if P0; P1 are �nite;then for any R⊆P0 × P1; (P0; P1; R) can be coded.

Proof. SC contains parameters c0; d0; c1; d1 coding P0; P1 and further parameters for therelation R. Suppose that P0 = {p0; : : : ; pn} and P1 = {q0; : : : ; qm}. First we assume that,in addition,

pi ∨ qj¡1 for all i; j: (4)

We will reduce the general case to this.As in the proof of Lemma 2.6(ii) there are g; h such that

E(g; h) = E({pi ∨ qj:Rpiqj}):

We claim that

Rpiqj ⇔ ∃z ∈ E(g; h)− {1}[pi ∨ qj6z]:


For the direction from left to right, simply let z= pi ∨ qj. For the other direction, sup-pose that the right-hand side holds via z¡1. By Lemma 2.2, z= inf{z∨ pr ∨ qs:Rprqs}.But, if not Rpiqj, then z∨ pr ∨ qs= 1 for each pair pr ; qs in R, since (i; j) 6=(r; s) andtherefore pi ∨ qr = 1 or pj ∨ qs= 1. This contradicts z¡1.Now, let

’rel(x; y; c0; d0; c1; d1; g; h)⇔’P(x; c0; d0) & ’P(y; c1; d1)

& ∃z¡1[x; y6z & z ∈ E(g; h)]:

Then in this special case each R ⊆ P0×P1 can be coded via ’rel.To remove the restriction (4) we imposed, we interpolate with a third EN-set. By

Proposition 3.2, there is an EN-set C0; : : : ; Cn such that, for all k6n, pi ∨ Ck¡1 andqj ∨ Ck¡1 (i6n; j6m). Let F : P0 7→ {C0; : : : ; Cn} be a bijection. Consider the relation Rgiven by RCkqj ⇔ R[F−1(Ck)qj]. Both F ⊆ P0×{C0; : : : ; Cn} and R ⊆ {C0; : : : ; Cn}×P1can be coded by parameters via ’rel. Then R=FR can be coded via the followingformula (think of z as F(x)):

’rel(x; y; �p)⇔∃z[’rel(x; z; c0; d0; c2; d2; g0; h0)& ’rel(z; y; c2; d2; c1; d1; g1; h1)];

where c2; d2 are parameters coding the auxiliary EN-set and �p consists of all 10parameters.

The following proof is somewhat more elementary than the previously known ones,because it only uses the exact pair Theorem 2.5, the technique of the Sacks splittingtheorem and Theorem 3.1 (i.e. the technique of the Lachlan Non-Diamond Theorem)as the recursion-theoretic ingredients.

Theorem 3.5 (Ambos-Spies et al. [3]). Th(Rwtt) is undecidable.

Proof. We use the usual general framework to prove undecidability of theories indi-rectly (see e.g. [5]). The class C of �nite directed graphs has a hereditarily undecidabletheory. Using Proposition 3.4, C can be uniformly coded in Rwtt. Hence Th(Rwtt) isundecidable.

4. Interpreting a copy of (N;+;×)

In the following, it is vital to keep in mind Convention 1:1.

Theorem 4.1. A copy of (N;+;×) can be interpreted in Rwtt without parameters.

We will use �nite EN-sets to represent numbers. The scheme SC from Proposition3.4 enables us to express by a �rst-order condition on parameters that EN-sets have


the same cardinality, and also the arithmetical operations. In the end we face theharder problem to single out �nite EN-sets. (Note that, even if our examples were all�nite, there is no reason to believe that all the de�ned via the scheme for EN-sets inDe�nition 2.7 are �nite.)We introduce the formulas without parameters to code (N;+;×). We use formulas

’num(�x); ’=(�x; y); ’+(�x; �y; �z) and ’×(�x; �y; �z), where �w stands for a pair of variablesw0; w1 which represent an exact pair needed to code an EN-set. The formula ’num(�x)will be considered last, but of course it will imply the correctness condition for SP ,since �x is thought of as coding an EN-set.1. Equality: Let ’≡(�x; �y) be a formula expressing

∃C[C is bijection P �x 7→ P �y];

using the scheme SC from Proposition 3.4. By that proposition, if P�a and P�e are �nite,then

|P �a| = |P �e| ⇔ Rwtt |= ’≡( �a; �e):

2. The arithmetical operations: Let ’+(�x; �y; �z) be a formula expressing that P�z canbe partitioned into two sets of the same size as P �x and P�y, respectively,

∃ �u∃ �v[’≡( �x; �u) & ’≡( �y; �v) & P�z = P �u ∪ P �v &P �u ∩ P �v = ∅]:

It can easily be checked that, for �nite P�a; P�e; P�c

|P �a|+ |P �e| = |P �c| ⇔ Rwtt |= ’+( �a; �e; �c):

For the direction from left to right one uses that subsets of P�c are again EN-sets.For ’×(�x; �y; �z) we say in terms of de�nable projection maps that P�z has the same

size as the cartesian product P �x × P �y. Thus ’×(�x; �y; �z) expresses

∃C1∃C2 C1:P�z 7→ P �x onto & C2 : P�z 7→ P �y onto

& ∀a ∈ P �x ∀b ∈ P �y ∃!q ∈ P�z[C1(q) = a&C2(q) = b]:

Then, for �nite P �a; P�e; P�c

|P �a||P �e| = |P �c| ⇔ Rwtt |= ’×( �a; �e; �c):

4.1. Recognizing �niteness

To recognize in a �rst-order way that an EN-set coded by two parameters is �nite,the idea is to compare EN-sets to fragments of a uniformly de�nable subclass of the ID-sets. ID-sets are not as easy to construct as EN-sets, but a more involved construction


actually yields a u.c.e. in�nite ID-set

Z∗ = {ai: i ∈ N}:To specify the uniformly de�nable subclass of the class of ID-sets we will imposeconditions on parameters c; d coding Z =Zc;d which are satis�ed by Z∗ and imply that1. when x ranges through degrees 6c; d , then |Z ∩ [0; x]| assumes all �nite cardinali-ties

2. if |P|= |Z ∩ [0; x]|, x6c; d , then a bijection between the two sets can be uniformlyde�ned.

ID-sets Z satisfying the conditions to be formulated will be called good. For the specialgood ID-set Z∗=Z∗

c;d , Z∗∩ [0; x] is �nite for x6c; d . The formula ’num implies about

P that for each good Zc;d , a bijection between P and some set Z ∩ [0; x], x6c; d , exist.The set Z∗ is obtained applying a rather hard theorem of Ambos-Spies and Soare

[4]. To ensure property 2. above, one has to make all the degrees ai low. An easierresult in Lempp and Nies [10] could also be used, but this has the disadvantage thatthe actual construction needs to be extended by adding new requirements to make thedegrees ai low.

Main Lemma 4.2 (Ambos-Spies and Soare [4]). There exists a u.c.e. sequence (Ai)i∈Nsuch that each Ai is low; Ai; Aj form a T -minimal pair for i 6= j and; where ai=degwtt(Ai); nd[0; ai] for each i. Thus Z∗= {ai} is an ID-set.

Proof. Recall that noncomputable c.e. set C is non-bounding if there is no minimalpair A; B such that A; B6TC. This de�nition makes sense also for wtt-reducibility.Clearly, C is wtt-non-bounding i� nd[0; d ] for each nonzero d6c=degwtt(C).In Ambos-Spies et al. [3, Lemma 6] it is proved that each non-bounding C is also

wtt-non-bounding. From Ambos-Spies and Soare [4] one obtains a u.c.e. sequence (Ai)such that each Ai is T -non-bounding and Ai; Aj form a T -minimal pair for i 6= j. Sincethere is a uniform construction to produce from a given c.e. set A a low set A suchthat A is non-computable if A is [18], we can assume that each set Ai is low.

De�nition 4.3. An ID-set Z de�ned from parameters c; d is good if(i) ∀x6c; d(Z * [0; x])(ii) ∀x6c; d ∃P

{〈u; C〉: u6C & u ∈ Z ∩ [0; x] & C ∈ P} (5)

is a bijection between Z ∩ [0; x] and P.Being good can be expressed by a �rst-order condition on c; d . Moreover, (i) implies

that Z is in�nite: else x= sup Z is below c; d , contrary to (i).We will prove that any u.c.e ID-set Z of low wtt-degrees is good, when de�ned from

an exact pair for the �03-ideal generated by Z . In particular the set Z∗= {ai} from theMain Lemma 4:2 is good. Assuming this fact, we now give a �rst order condition onparameters expressing �niteness of an EN-set P.


Lemma 4.4. P is �nite ⇔ ∀a; b[Za;b good ⇒∃x6a; b∃P[the map (5) is a bijection & ∃C C is bijection P ↔ P]]:

Proof. For the direction from left to right, assume that P is �nite. Because good ID-setsare in�nite, we can choose F ⊆Z such that |F |= |P|. If x= sup F (with the conventionsup ∅= 0), then by (ii) of Lemma 2.1 F = [0; x]∩Z . Choose P satisfying (5). ByProposition 3.4, a bijection P↔P can be coded via SC .For the other direction, let a; b be an exact pair coding the set Z∗ obtained from

the Main Lemma 4:2. If x6a; b, then x6a0 ∨ : : : ∨ an for some n. By Lemma 2.1,ak ∧ x= 0 for all k¿n, so Z ∩ [0; x] is �nite. Thus P is �nite.

Finally, we supply the proof that any in�nite u.c.e. ID-set Z of low wtt-degrees isgood. Let Z be such a set, coded by an exact pair a; b. By a similar argument asabove, Z ∩ [0; x] is �nite for any x6a; b. Since all degrees in Z are low and the �niteEN-sets are closed downwards, it is now su�cient to prove the following.

Lemma 4.5. Suppose that a0; : : : ; an are low pairwise incomparable degrees in Rwtt.Then there is an EN-set C0; : : : ; Cn such that

ai6Cj ⇔ i = j:

Proof. Choose c.e. sets Ai ∈ ai. We construct c.e. sets Vj such that the statement ofthe lemma holds with Cj =degwtt(Aj ⊕Vj). Clearly ai6Ci. To ensure aiCj for i 6= j,we meet the requirements

Ne; i; j: Ai 6= [e]Aj⊕Vj (i 6= j);

by the same strategy as in the proof of Proposition 3.2: refrain from changing Vj tilla permanent disagreement occurs. We will de�ne some priority listing (Rk)k ∈N of allthe requirements. If Rk is Ne; i; j let

length(k; s) = min{x: ∀y ¡ x Ai(y) = [e]Aj⊕Vj (y)[s]

and let r(k; s)= max{[e]s(y): y¡length(k; s)}.To achieve Cj ∨ Cj′ = 1 (j′ 6= j) as in Proposition 3.2 we ensure that K =Vj ∪Vj′ . For

nd[Cj; 1], we make each Aj ⊕Vj low and apply Theorem 3.1. Lowness is achieved bythe side e�ects of the “pseudolowness requirements”

Le; j: ∃∞s {e}Aj⊕Vj (e)[s] is de�ned ⇒ {e}Aj⊕Vj (e) converges:

While Le; j may fail to be met, it will produce enough restraint to ensure (Aj ⊕Vj)′ ≡T ∅′.We use a standard technique introduced by Robinson. By the recursion theorem, wecan assume that the sets V0; : : : ; Vn with speci�c enumerations are given (see commentat the end). Since each set Ai (i6n) is low, the following property of e; j and a stagenumber s can be checked with an oracle ∅′:

∃s¿s[{e}Aj⊕Vj (e)[s] is de�ned via an Aj-correct computation]: (6)


By the Limit Lemma [18] we choose a computable function g(s; e; j; t) such thatlimt g(s; e; j; t) exists, has value 0 or 1, and the limit equals 1 i� (6) holds. Let (Rk)be some priority listing of all the requirements.

4.2. Construction

At Stage 0 initialize all the lowness requirements.Stage s+1: First determine the restraint r(k; s) for all k¡s such that Rk is a lowness

requirement Le; j. Let s¡s be greatest such that Rk was initialized at s. If {e}Aj⊕Vj (e)[s]is unde�ned, let r(k; s)= 0. Else let u be the use of this computation and �nd the leastt¿s such that either(a) Aj; t+1|u 6=Aj; t |u, or(b) g(s; e; i; t)= 1.Since (6) is equivalent to limt g(s; e; j; t)= 1 and the computation at s seems to providea witness for (6), one of the two cases has to apply. In Case (a) let r(k; s)= 0, whilein Case (b) r(k; s)= u.Now, if Ks=Ks+1 terminate stage s + 1 here. Else, say y is the unique element in

Ks+1 −Ks. Determine the minimal k such that y¡r(k; s). If k fails to exist enumeratey into all the sets Vj. Else let j be the number such that Rk =Le; j or Rk =Ne; i; j forsome e; i. Enumerate y into Vj′ , for each j′ 6= j. Initialize all the lowness requirementsR′k , k

′¿k. This completes the description of the construction.

Lemma 4.6. Let k¿0.(i) If Rk is Ne; i; j ; then the requirement Rk is met.(ii) r(k)= lims r(k; s) exists and is �nite.

Proof. Assume the lemma holds for all h¡k. Choose a stage s0 such that for all h¡k,r(h; s0) has reached the limit, and K does not change below maxh¡k r(h) at any stages¿s0.If Rk is Ne; i; j, we can prove (i) and (ii) as in Proposition 3.2. In particular, if

Ai= [e]Aj⊕Vj , then one can obtain a wtt reduction procedure of Ai to Aj, contrary tothe assumption that ai ; aj are incomparable.Now suppose that Rk is Le; j. We have to show that lims r(k; s) is �nite. Let s be

the greatest stage where Rk is initialized (necessarily s6s0), and pick s¿s0 whereg(s; e; j; s) has reached its limit. If the limit is 0, then r(k; t)= 0 for all t¿s. Else,by (6) and the de�nition of g there is a least stage t¿s such that {e}Aj ⊕ Vj (e)[t] isde�ned via an Aj-correct computation with use u . Then at stage t we de�ne r(k; t)= u.Since Rk is not initialized at stages ¿s, the computation {e}Aj ⊕ Vj (e)[t] is preserved.So r(k; s)= u for all s¿t.

Lemma 4.7. Aj ⊕Vj is low for each j6n.

Proof. Given e, we have to determine with a ∅′-oracle whether {e}Aj ⊕ Vj (e) converges.Let k be such that Rk is Le; j. Note that, in the proof of the preceding lemma, we can


determine s using a ∅′-oracle. Then, by (6),limt

g(s; e; j; t) = 0 ⇒ {e}Aj⊕Vj (e) diverges

and by the argument above,

limt

g(s; e; j; t) = 1⇒{e}Aj⊕Vj (e) converges:

The use of the recursion theorem deserves a comment: We are given some c.e. setsV0; : : : ; Vn via a partial recursive enumeration function which maps s to a strong indexfor V0 ⊕ : : : ⊕ Vn[s]. From this the construction produces a similar enumeration forsets V0; : : : ; Vn. By the recursion theorem, there must be such that = , and inparticular Vj =Vj for j6n. The function g actually contains an extra argument, namelyan index for , and in the discussion above we assume that this extra argument is anindex such that = .

References

[1] K. Ambos-Spies, An extension of the nondiamond thoerem in classical and �-recursion theory, J.Symbolic Logic 49 (1984) 586–607.

[2] K. Ambos-Spies, P.A. Fejer, S. Lempp, M. Lerman, Decidability of the two-quanti�er theory of therecursively enumerable weak truth-table degrees and other distributive upper semi-lattices, J. SymbolicLogic 61 (3) (1996) 880–905.

[3] K. Ambos-Spies, A. Nies, R.A. Shore, The theory of the recursively enumerable weak truth-table degreesis undecidable, J. Symbolic Logic 57 (1992) 864–874.

[4] R.I. Ambos-Spies, K. Soare, The recursively enumerable degrees have in�nitely many one types, Ann.Pure Appl. Logic 44 (1989) 1–23.

[5] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer, Berlin, 1981.[6] R. Dowey, A. Nies, G. LaForte, Enumerable sets and quasireducibility, Ann. Pure Appl. Logic 95

(1998) 1–35.[7] L.A. Harrington, A. Nies, Coding in the lattice of enumerable sets, Adv. Math. 133 (1998) 133–162.[8] W. Hodges, Model Theory, Enzyklopedia of Mathematics, Cambridge University Press, Bangkok, 1993.[9] A.H. Lachlan, Embedding nondistributive lattices in the recursively enumerable degress, in: W. Hodges

(Ed.) (Ed.), Conference in Mathematical Logic, London, 1970, Lecture Notes in Mathematics, Vol.255, Springer, Heidelberg, 1972, pp. 149–177.

[10] S. Lempp, A Nies, Undecidability of the 4-quanti�er theory for the recursively enumerable turing andwtt degrees, J. Symbolic Logic 60 (1995) 1118–1135.

[11] A. Nies, De�nability and undecidability in recursion theoretic semilattices, Ph.D. Thesis, Universit�atHeidelberg, 1992.

[12] A. Nies, The last question on recursively enumerable many-one degrees, Algebra Logika 33(5) (1995)550–563. (English Translation: Consultants Bureau, NY, July 1995).

[13] A. Nies, Parameter de�nable subsets of the computably enumerable degrees, to appear.[14] A. Nies, R.A. Shore, Interpreting true arithmetic in the theory of the r. e. truth table degrees, Ann. Pure

Appl. Logic 75 (1995) 269–311.[15] A. Nies, R. Shore, T. Slaman, Interpretability and de�nability in the recursively enumerable

turing-degrees, Proc. London Math. Soc. 3 (77) (1998) 241–291.[16] R.A. Shore, The theory of the degrees below 0′, J. London Math. Soc. 24 (1981) 1–14.[17] T.A. Slaman, W.H. Woodin, De�nability in the enumeration degrees, Arch. Math. Logic 36 (1997)

255–267.[18] R.I. Soare, Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Omega

Series, Springer, Heidelberg, 1987.

Interpreting N in the computably enumerable weak truth table degrees

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