Arch. Math. Logic (1995) 34:183-196 Aeehix,e for Mathematical Logic 9 Springer-Verlag 1995 On injective enumerability of recursively enumerable classes of cofinite sets Stephan Wehner* Department of Mathematics and Statistics, Simon Fraser University, Bumaby, Canada V5A 1S6 (e-mail: wehner@~fu.ca) Received January 13, 1994 Abstract To date the problem of finding a general characterization of injective enumerability of recursively enumerable (r.e,) classes of r.e. sets has proved intractable. This paper investigates the problem for r.e. classes of cofinite sets. We state a suitable criterion for r.e. classes ~ such that there is a bound n C w with Iw - A I _< n for all A c W. On the other hand an example is constructed which shows that Lachlan's condition (F) does not imply injective enumerability for r.e. classes of cofinite sets. We also look at a certain embeddability property and show that it is equivalent with injective enumerability for certain classes of cofinite sets. At the end we present a reformulation of property (F). I Introduction To date the problem of finding a general characterization of injective enumerability of recursively enumerable (r,e.) classes of r.e. sets has proved intractable. There are several sufficient criteria known and only one necessary criterion is published. Most sufficient criteria for a recursively enumerable class W to have an injective enumeration are of the form: there is an injectively enumerable subclass ~ of ~ (of a certain kind) such that every finite subset of a member of ~ - ~ has infinitely many extensions in ~ [K, KD ch. 4, M]. In [L,LC] the condition (E) is shown to imply injective enumerabitity. A class ~c~ satisfies (E) if one can uniformly pass from finitely many recursively enumerable sets V 1 , . . . , Vn to an enumeration of ~ - { V l , . . . , V, }. No combination of these ideas promises to yield a necessary condition. The only necessary criterion to be found in the literature is the condition (F), see [L]. At the end of this paper we show that a recursively enumerable class W satisfies (F) if and only if * Thank you for technical support, WolfgangEppler, for intellectual support, Alistair Lachlan, and for proof-reading, Martin Kummer. Thanks also to the anonymousreferee 184 S. Wehner (G) there exists an injectively enumerable class c ~ C W such that every finite set has the same number of extensions in ~ as in ~ . Not many examples of r.e. classes which have no injective enumeration are known. We want to mention two here. The first one we mention for its simplicity: let K be any recursively enumerable, non-recursive set. Then the class ~K : = { { 2 X , 2 X + I } : x EK}U({2x},{2x+I}:x ~K} is recursively enumerable because K is, and ~ is not injectively enumerable because co - K is not recursively enumerable. The second example is one with a rich structure: Martin Kummer [K] constructs an r.e. class ~ " of finite functions such that for every i C co at most one finite function d with d(0) = i is not in ~ - (this implies that ~ " is injectively enumerable) and such that ~ 1 U .~" is recursively enumerable but not injectively (,-~1 is the class of recursive functions). This example shows that structure is not the critical property of injectively enumerable classes. As Martin Kummer writes, one cannot hope for "static" necessary conditions. Also, his example shows that (F) is not a sufficient criterion, because clearly the class ~ 1 U .~" satisfies (G) (choose ~ = ~ ' ) . This paper investigates the problem of characterizing injective enumerability for recursively enumerable classes of cofinite sets. Section 3 deals with classes such that there is a bound n c co with ~ ( W ) ; ~ ( W ) means that Iw AI <_ n for all A E ~ . For classes with ~ ( W ) a "dynamic" property is stated which characterizes the injectively enumerable classes of this kind. It connects injective enumerability and a certain simplicity-notion. For each number n we show that a very simple necessary condition of injectively enumerable classes is also sufficient for the injective enumerability for classes W with , ~ ( ~ ) . On the other hand, in Sect. 4 an example is constructed which shows that the condition (F) does not imply injective enumerability for r.e. classes of cofinite sets. In Sect. 5 we consider a class which is injectively enumerable but which does not fall under any of the known sufficient criteria. However, we look at a certain embeddability property and find it to explain (in a weak sense) the injective enumerability of this class (and many other classes of its kind). But we also show that it does not yield a necessary condition in general. 2 Definitions We denote the set of natural numbers by w. Let D = {D i : i C co} be a canonical enumeration of all finite sets of natural numbers. A class W C 2 ~ is called recursively enumerable if its members are uniformly recursively enumerable; this means there is a partial recursive binary function c~ such that {range(An.cffi, n)) : i E co} = ~ . Such a function c~ is called an enumeration of and c~i :=range(An.c~(i, n)), the i-th set of the enumeration. An enumeration is called injective if for different indices i , j c w the i-th and j-th set of the enumeration are different. Let {., .} : w 2 ~ w be a recursive bijection and let qa be a G6del-Numbering of the class of partial recursive functions. Define Injective enumerability:classes of cofinite sets 185 Wi := range(T/) 3Q~e) : : W~pe(i) ~(~) := {/2~e)" i E w}, where W~e(i) := 0 if ~e(i) is not defined. J2 enumerates all enumerations of recursively enumerable classes. Given an enumeration o f an r.e. set A we denote by As the subset of A enumerated after performing s steps of the enumeration. For a set A C_ w and a number x the set A A { 0 , . . . , x - 1} is denoted by A Ix; when we write x + A we mean the set {x + a : a E A}. By convention we let max(0) = - 1. 3 B o u n d e d classes of cofinite sets Recall that ~ ( W ) means that Iw - A[ < n for all A E W. In this section we characterize injective enumerability for the recursively enumerable classes such that ~ ( W ) holds for some bounding number n. First we look at classes ~ with ~(~). It turns out that understanding injective enumerability for these classes is crucial for the more general case. Observe the following: Proposition 3.1 Let ~ ~(~). be an infinite recursively enumerable class such that Then 1. ~ is injectively enumerable if and only if ~ - {w} is recursively enumerable. 2. ~ - {~o} is recursively enumerable if and only if ~ - {w} has an infinite r.e. subclass. Proof The 'only if'-part of the second statement is trivial. The if-part can be seen as follows. Let S ,~ be an infinite r.e. subclass of W - {w}. From a pair (e, i) we can effectively find an index o f a recursive enumeration of the class 9~,i 9 defined by: S~ i f / E We ~9P U {We} otherwise. Now let V be an r.e. set such that W = {We : e E V}. Then ~ because it is equal to U {S~,i : e E V, i E ~o}. The first statement follows from the height-criterion 1 [P]: {w} is r.e. T h e o r e m 3.2 lf for an r.e. class Jd" C ~ Y there is a partial recursive one-place function h (called height-function), such that 1. (Vi)(h(i) ~r (~tA E .~'~w~)(Oi C A)) 2. (Vi,j)([h(i),h(j) ~ and Di C_ Dj] ~ h(i) <_h(j)) 3. (Vi)(h(i) ~ (~j)(h(j) l,Di c_ Dj, and h(i) < h(j))) 4. (VA E .J~C)({h(i) :Di C_A} is finite) I A class with a height-function also has the property (E), see [L] 186 S. Wehner then ~ has an injective enumeration. If an r.e. class ~ satisfying ~ ( ~ ) is infinite then for every i E w the set { 0 , . . . , i } has an extension different from w in W and the function h(i) := Izx.(x ([ Di) is a height-function for W - {w}. Therefore recursive enumerability of W - {w} implies injective enumerability of ~ . [] An r.e. class ~ with ~ ( ~ ) is determined by the set {x " w - {x} E ~ } and whether w is a member of ~ . So the question of characterizing injective enumerability of such classes is a question about sets of natural numbers. And because of the observation above the question is: For which sets of natural numbers A such that the class {w - {x} : x E A}U{w} is recursively enumerable 2, is the class ~ := {w - { x } : x E A} also recursively enumerable? An answer can be given in terms of ]--simple sets: Detinition 3.3 A set A C ~o is called ],-simple if and only if A is recursively enumerable and there is no disjoint weak array which covers w - A , and each of whose members intersects w - A in exactly one point. T h e o r e m 3.4 For a set A C w such that {w - {x} : x E A} U {w} is recursively enumerable, the class ~a is recursively enumerable if and only if there is a subset of A whose complement is r.e. but not Z-simple. Proof To prove the if-part, suppose that B C A is a co-r.e, subset whose complement is not 1-simple. Let (Fi)iE w be a disjoint weak array witnessing that w - B is not 1-simple. Then {w {x} "x E B} is enumerated by (Tn)nE~ where 7 is defined by "yn : = ( w - B ) u U F i . i4n By the second part of Proposition 3.1 it follows that ~A is recursively enumerable. Let us turn to the "only if'-part. Suppose {w - {x} : x E A} is recursively enumerable. Then by Proposition 3.1 this class has an injective enumeration, call it ~b. We want to show that there is a co-r.e, subset of A whose complement is not 1-simple. We carry out the following construction while simultaneously effectively enumerating the sets ~i (i E w). Prior to stage s we will have chosen distinct numbers fo,. 9 9,fi-1. Stage s. Part 1. Advance the enumeration of the sets ~i (i E w) until the following is true: for each i < s, if bi,s denotes the least number not yet in ~ , then bi,s E ~fj for each j < s,j y! i. Part 2. Advance the enumeration of the sets ~i (i E w) until k ~( Y0~, : i < s} is found such that {n : (3i < s)(n <_ bi,s)} C ~k. Setfs = k. 2 In fact, the sets A for which {w - {x} :x E A} U {w} is recursively enumerable are precisely the Z72-setsof the Arithmetical Hierarchy Injective enumerability:classes of cofinite sets 187 End of construction. For each i E co let bi denote the unique number such that % = w - {bi}. It is clear that for each i, bi,s = bi for all sufficiently large s. Now we define B := {bi : i E w} and Fi := {bi,i+l, bi,i+2,...}. Clearly B C_ a and it is easy to check that that the sets Fi are pairwise disjoint and cover B, each single set F] intersecting B exactly once. Further, w - B is r.e. because n is in B if and only if there exists s > n such that n ~ {bo,s,..., bs-l,s}. This is enough. [] Let H be a hyperhypersimple set. Then the class ~ n := {~o - {x} : x H } tO {w} is recursively enumerable but not injectively, because no superset of H is 1-simple. Note, however, that we can enumerate ~ / 4 - ~ for every finite subclass ~ of ~ 4 n not including ~o. We use this observation to prove the following T h e o r e m 3.5 For every number n > 0 there is an r.e. class ~ C_ 2 ~ such that ' ~ ( ~ ) holds, ~ has no injective enumeration but every subclass ~ t C ~ with [~ _ ~ [ < 2 n- 1 is recursively enumerable. Proof Given a number n > 0 we choose a hyperhypersimple set H and define ~:={(n-1)+A:Ac,~n}U{w-B'B C_ {x C w : x <n-l}}. is not injectively enumerable because ~/~u is not. On the other hand, any one of the 2 n - 1 sets w - B, where B C_ {x E o; : x < n - 1}, can be used as an escape set to enumerate ~ . [] We digress shortly to address a question of Julia Knight. She asked me, whether a class of subsets of the natural numbers, which is recursively enumerable in every non-recursive Turing degree is recursively enumerable. The answer to this question is yes, because if a set is r.e. in 0 t and in a hyperimmune-free degree, then it is r.e. 3 Also it is possible to construct a non-recursive degree m such that every set which is recursively enumerable in m is recursively enumerable. A related question is, whether there are two r.e. non-recursive degrees such that every class which is recursively enumerable in these two degrees is recursively enumerable. This problem remains open but we obtain the following result: T h e o r e m 3.6 Suppose that a and b are two recursively enumerable degrees such that if a class o f sets o f natural numbers is recursively enumerable in both a and in b, then this class is recursively enumerable. Then a and b cannot both be high. The relevance of this theorem lies in the fact that there are two high recursively enumerable degrees, such that any set which is recursive in both, is recursive [LM]. A slight modification of the underlying construction yields two high r.e. degrees, such that any set, which is recursively enumerable in both, is recursively enumerable. [Yi] 3 Thanks to Martin Kummer for pointing this out 188 S. W e h n e r Proof Given two high r.e. degrees a, b let Ha, Hb be hyperhypersimple sets in a, b, respectively. Let H be the intersection of Ha and Hb. Now ~'~H is recursively enumerable. We see that w - Ha and w - Hb is recursively enumerable in a and b, respectively. We also see that aJ - H D w - Ha, w - Hb. Therefore the class ~'~t/ - {w} has infinite recursively enumerable subclasses relative to a and b. Note that part 2 of Proposition 3.1 can be relativized. Therefore J ~ u {w} is recursively enumerable in a and in b. However, the remark immediately preceding Theorem 3.5 shows that ~ u is not recursively enumerable, because H is hyperhypersimple. [] Our next theorem characterizes the r.e. classes W with ~ ( W ) which are injectively enumerable. The general approach is to work with subclasses of W which admit a height-function. The bound hinted at in Theorem 3.5 turns out to be sharp: T h e o r e m 3.7 Let W be an infinite r.e. class such that ~ n ( W ) . Then C is injectively enumerable if and only if every W ' c W such that ]W - Wt[ < 2 n-1 is recursively enumerable. Proof Necessity is obvious. To prove sufficiency, suppose a class W is given, such that the hypothesis holds, i.e. Iw -AI < n for all A E W and every W t C W with ]W - W I1 < 2 n-1 is recursively enumerable. Define the following relation ___ on finite sets a, ~- C w : O" [ - T :r a : T [max(a)+l . Given a class 5 ~ C_ 2 ~ we say that a finite set a C_ ~ is .~2-infinite-branching if for every a c w there is a set B E ~ such that a _ w - B and min(w - (B t2 a)) > a. A set is minimal ~-infinite-branching if it is 5~-infinite-hranching but contains no proper subset which is 5~-infinite-branching. The following statement can be easily proved by induction: L e m m a 3.8 Let ~ C 2 ~ be a class such that ~ m ( ~ ) . Then there are only finitely many minimal ~-infinite-branching sets. Also the complement o f almost every set in ~ contains a c~-infinite-branching set, and .~-infinite-branching sets have cardinality less than m. The key to finding the injective enumeration of W is provided by: L e m m a 3.9 For any minimal W-infinite-branching set a the class ~ W : a E w - A} is injectively enumerable. := {A C Proof It is enough to demonstrate that := % - - .} is recursively enumerable; a simple application of the height criterion (namely using the height function #x.x > max(a) and x r D i and D i C w - - a ) then shows the injective enumerability of ~ . By the condition on W we can enumerate Injective enumerability:classes of cofinite sets ~ " := ~ - 189 {co-M : M C_ a} for there are at most 2 n-1 subsets of a. Let us look at sets a - {n} for members n of a. These sets are not infinite-branching, because a is minimal infinitebranching and so there must be a finite set F > max(a) such that (VA E 7~")[(3n)(a - {n} F- co - A ) ~ (3x E f ) ( x CA)] With the help of such a set F we can enumerate ~ as follows: Let 7 be an enumeration of ~ " N {A : { 0 , . . . , max(a)} - a C_ A}. Define n ifo'N')'n,s=0 m if s is least such that a N %,s ~ 0 and where <m, t) is least such that (F U % , , - 1 ) C_ 7m,t fi(n) otherwise fs+l(n) := and /3. := Us>o %(.),~. N o w / ~ enumerates ~ . [] To complete the proof of the theorem, let us look at m := max{x : (3a)(a is minimal ~-infinite-branching and x E a)}. If m = max{0} = - 1 then 0 is the only minimal ~-infinite-branching set, and Lemma 3.9 shows that ~ is injectively enumerable. So suppose m > 0. Below we will show that ~c~. is recursively enumerable, where ~ * denotes ~* := {AE ~:co-Ahasanelement >m and a ~-infinite-branching subset}. Since the function )~i.(px > m.x • Di and (3.4 E ~C*)(Di C A)) is a height-function for ~ * , it follows that ~c~. is injectively enumerable. By Lemma 3.8 the difference ~ - ~c~, is finite. Hence the class ~ is injectively enumerable. Why is ~ * recursively enumerable? For every minimal ~-infinite-branching set a Lemma 3.9 provides us with an enumeration of ~c.:={AE ~:w-Ahasanelement >m andarw-A}. Using these enumerations we can avoid enumerating sets whose complements have no ~c~-infinite-branching subset. For the purposes of the followlng construction we suppose given a list L of the minimal ~6%infinite-branching sets. Further, for each a in L we suppose given an index of an injective enumeration 0 ~ of ~ * . Finally, given an enumeration %bof ~ , we define: 190 S. Wehner ~bi Bi,x := { if (Vs)({0,...,m} and x ft ~i,s ) - - ffYi,s has an infinite-branching subset if s is the least such that ~i,, contains x or meets every member of L, ~r is first in L such that ~r A ~bi,s-1 and (m,t) is least such that 0~, t ~_ ~bi,s-1. Then {Bi,x : i , x E w , x > m} = ~ * and Bi,x is recursively enumerable uniformly in i and x. [] 4 A counterexample for Lachlan's condition (F) Consider the following property of a class W of r.e. sets: (G) There exists an injective enumerable class ,.~ _ W such that every finite set has the same number of extensions in ._~ as in W. The property (G) is clearly necessary for the injective enumerability of W. Further, it is clear that (G) implies Alistair Lachlan's property (F) 4, which is the strongest necessary condition for injective enumerability to be found in the literature. The last section presents a proof that (F) and (G) are in fact equivalent. Martin Kummer [K] gives an examPle of an r.e. class which contains all recursive functions and an injectively enumerable class of finite functions, has no injective enumeration but satisfies (G). In this section we construct an r.e. class of cofinite sets which is not injectively enumerable and which satisfies (G). An inspection of the proof of Theorem 3.7 shows that for classes W with ~ (W), the property (G) implies injective enumerability. Lemma 4.1 Uniformly in a recursive enumeration ~b we can injectively enumeI{A : I• - A[ = 1} - ~ , J < 1 and ~ does rate a class ~ such that ~ ( ~ ) , not injectively enumerate ~ U {w}. With this lemma the following theorem can be proved easily: Theorem 4.2 There exist r.e. classes ~ ~_ . ~ of cofinite sets such that ~ is injectively enumerable, ._A~ is not, and every finite set which has an extension in ~ has infinitely many extensions in ~ . Proof Let ~ / denote the class ,~s~u~ obtained from Lemma 4.1. Let D (i) := {0, 2, 4 , . . . , 2i, 2i + 1}. Define ._~ := {O (i~ U ((1 +max(D@)) +X) : i E w and X E ~,,- U {~}}, d ~ := {D@ U ((1 +max(D(i)))+X) : i E w andX E ~ / } . From the properties of the classes ,5~au~, i E w it follows that ~ conclusion of the theorem. [] 4 See Sect.6 for a definition satisfies the Injective enumerability:classes of cofinite sets 191 Proof of Lemma 4.1. Given the recursive enumeration ~b we construct an injective enumeration/3 such that fl enumerates ~ for which the conclusion of the lemma holds. The class ~ will be equal to {A : [w - A [ = 1} or this class reduced by one member or this class extended by one element, namely ~. For every n C w we rely on the n-strategy to satisfy {~bm : m > n} ~ . _ ~ - {w}. The strategy consists in choosing a number m > n and leaving ~b,~ out of ~.~, if it is not w. If the strategy is successful, ~b does not injectively enumerate ~ U {w}. However, the n-strategy has an effect on ~ only if n is the minimal ~b-index of a~. Let f be a recursive binary function such that 1. f ( x , s) is increasing in x and non-decreasing in s, 2. f ( 0 , s) = 0, 3. limsf(X,S) = cx~ if and only if ~by = w for some y < x. Using f we construct a binary partial recursive function 9. The meaning of the function 9 is that/3 n will be defined as co - {lims 9(n, s)}. The meaning of the function f is that at stage s + 1 the n-strategy may assign 9(x,s + 1) a value different from 9(x,s) only i f f ( n - 1) < x < f ( n ) . We say that the n-strategy requires attention at stage s + 1 if there are numbers in,in such that each of the following holds. 1. 2. 3. 4. f ( n - 1,s) < f ( n , s ~- 1) and (Vx < n)[f(x,s) = f ( x , s + 1)] in = #k > n.(min(w - ~bt,s) > m a x { 9 ( x , s ) :x < f ( n - 1,s)}) f ( n - 1,s) < A < f ( n , s ) and 9(in,s) ~= min(aJ - ~bi,,s) max{9(x,s ) : f ( n - 1,s) < x < g(jn,S)} <_ min(w - (~i,,s tA {gqn, S)})). Stage s + 1. Case 1. s + 1 is odd. Let n , j be the least such that the n-strategy requires attention, witnessed byjn = j . If there is such a number n then set 9(J, s + l ) = s + 1, let i be least such that 9(i,s) T and set 9(i,s + 1) = 9(j,s). Case 2. s + 1 is even. Let 9(j, s + 1) = min(w - rg(Ax.g(x, s))) where j is least such that g(/, s) T. For all k, if 9(k,s) J. and 9(k,s + 1) has not been set by the previous instructions, then let 9(k, s + 1) = 9(k, s). End of construction. Define fin "'~ {X " (~S > X)(9(n,s ) .~4 X)} 192 S. Wehner for all indices n and let o ~ = {/3n " n C w}. T h e n / 7 is defined uniformly in and we verify the following claims. Claim 1. For ~ we have ~1(.5~). I f g ( n , s + 1) : / g ( n , s ) I then 9(n + 1,s + 1) = s + 1 so that/3n is either the set {lims g(n, s)} or the set w depending on whether the limit exists or not. co - Claim 2. Suppose ~ enumerates w. Let no be the least index such that ~bno = w. 1. For every m there is a stage s such that if the value of 9(m) is changed at a stage t > s then no receives attention at stage t. 2. In all sufficiently large stages s at which no receives attention, ino has the same value. Let f(no - 1) denote limsf(no - 1,s). Let s be a stage such that 9 ( x , s ) .[ for all x <_ m , f ( n o - 1), none of 0 , . . . , no - 1 receive attention after stage s and f(n0, s) > m. N o w s satisfies 1. The values of 9(x) for numbers x < f(no - 1) do not change after stage s and so the final value of ino is given by (#k > n)(min(w - ~Pk) > max{g(x) : x < f ( n 0 - 1)}), where we let min(0) = c~. Claim 3. If/3 enumerates w then so does ~. Suppose/3, = w. Then the value of g(n) is changed in infinitely many stages. In every such stage some m < n receives attention and one m < n must receive attention infinitely often. For mo, the least such m, it follows that ~bmo = co. Claim 4. The enumeration/3 is injective. If g(n,s) is defined to be the number a at stage s, then g(nt, t) = a implies g(n, t) : / a for all t and all n ~ 5 / n . Therefore, if for n 5 / m , /3n = co - {a} and /3m = co - {b}, then a 5 / b . It remains to be shown that/3 enumerates co at most once. Suppose m0 is the least/3-index of co and ma > m0. By Claim 3 let n be the least ~-index of co and by Claim 2 let s be a stage such that if either g(mo) or 9(ml) is changed in a stage t > s then n receives attention in stage t. Furthermore let s be such that in has reached its final value in stage s. As/3m0 = co, the value of g(mo) is changed infinitely often. This means that the sequence of values that min(co - ~bin,s) takes on, diverges. Let s ~ > s be a stage such that n receives attention, g(mo, s), rain(co - ~i,,s) > ml and g(mo) is changed in stage s. Then either mm < f ( n - 1,s t) or g(mm) C ~i,,s. In both cases, the value of 9(ml) is not changed after stage s. Therefore/3ml :/co. Claim 5. ~b does not injectively enumerate c ~ U {co}. Suppose ~b enumerates c ~ tO {co} and that no is minimal such that ~bno = co. By the second part of Claim 2 let i be the final value of ino. If ~ i :/ co then co - (min(aJ - ~i)} ~ ~ . So ~)i = 03, and as i > no, ~b is not injective. Claim 6. If ~ does not enumerate w, then c ~ = {A : tco - A I = 1}. Injective enumerability:classes of cofinite sets 193 Fix a E co. Let n , s , x be such that 9 ( x , s ) > a , f ( n , s ) > x and there is a k such that 9 ( k , s ) = a. If at a stage t > s there is z such that 9 ( z , t ) = a and 9(z, t + 1) --/a then there is z ~ =/z such that 9(z', t + 1) = a, and at stage t some m < n must receive attention. As all numbers receive attention only finitely often, co - {a} E , ~ . By Claim 3 the conclusion follows. Claim 7. [ { a : [co -a[ = 1} - ~ [ < 1. By Claim 6 we may suppose that n is the least ~b - index of co. For a E co there are two cases. Case 1 is that there are only finitely many stages such that 9(m , s) = a and 9(m, s + 1) 5t a for some m E co. Case 2 is the negation of case 1. For case 1 it follows immediately that co - {a } E ~ . In case 2, n receives attention at almost all stages s such that 9 ( m , s ) = a , 9 ( m , s + 1) 5~ a for some number m. This is because there is a stage so and a number x such that 9(x,so) > a and f ( n , So) > x and none of 0 , . . . , n - 1 receive attention at a stage t > so. Now n is the only number for which it is possible to receive attention at a stage t > So such that 9 ( m , t ) = a and 9 ( m , t + 1) 5 t a for some m E co. By Claim 2, in has a final value. Therefore case 2 can occur for at most one number a. [] 5 Embedding injective enumerations in others There is a class, call it ~-~*, which is very similar to the class ~ constructed in Theorem 4.2 but which has an injective enumeration 5 ;"similar" means it is of the form (1) {D (i) U ((1 + max(D(i))) + X ) : i E w and X E c ~ / } where 6~1(~/), w E ~ / and [{A : [ c o - A I < 1 } - ~ / [ < 1 for all numbers i, and there is no straightforward way to describe which set is missing from each class ~ ( i ) . The class ~ * is obtained from the fact that one can uniformly construct, given an enumeration ~b, an injective enumeration of a class ~ such that ~ ( ~ r E ~r I{a : Ico-AI = 1} - ' - ~ w [ < 1 and ~b does not injectively enumerate ~,~r - {co}. This poses the question as to how to distinguish classes of this form which are injectively enumerable from those which are not. Also the injective enumerability of ~ * cannot be explained by any of the known sufficient criteria for injective enumerability. The class ~ * does not satisfy (E), because .J~* - {D(i)u((1 +max(D(i)))+co) : i E co} is not recursively enumerable. Also, no injectively enumerable subclass (which is not obtained from an injective enumeration of ,/~*) can be found that would fit the other criteria mentioned in the introduction. This poses the problem of finding a sufficient criterion for injective enumerability which covers these kinds of classes. 9 Both questions can be answered in terms o f embedding injective enumerations in injective enumerations of natural classes. Given any injective enumeration 7 and an injective recursive function 9, the enumeration "7(9) described by 3'(9)n := 79(n) (for all n E co) is clearly injective. In this case we say that "/(9) is embeddable in 7. Let W, W* denote the class enumerated by 3',7(9) 5 The proofs of this section are omitted for reasons of space 194 S. Wehner resp. If ~ is a natural class, whose injective enumerability is obvious, we can explain the injective enumerability of ~ * by the fact that g~ has many injective enumerations and one of them, namely 7 and some injective recursive function, namely g, together enumerate g~* injectively. The class oc~ := {A - {a} : A is co or an initial segment of co and a E A} U {w} is obviously injectively enumerable. It turns out that uniformly (modulo the information whether w C g~) in an injective enumeration of a class g~ with ~11(~) we can construct an injective enumeration 7 of o9~ and a recursive function g such that 7(g) injectively enumerates Z ~ - {w}. This tells us which classes of the form (1) such that ~11(o~/) and co c o~/ for all i E co have an injective enumeration: they are those for which the classes o ~ (i) are uniformly embeddable in an injective enumeration of o~ . H o w e v e r this approach does not yield an answer for the general question: one can prove that given an enumeration r a partial recursive function g and a number x one can injectively enumerate a class ~ = ~r such that . ~ ( ~ ) , x = min(co - A) for all A C g~ and such that if g is recursive then ~b(g) either does not enumerate co - {x }, repeats one of the sets in g~ or does not enumerate all sets in g~. The union i ,jEco then is injectively enumerable, but no injective enumeration of it is embeddable in an injective enumeration of an extension containing {A :]co - A I = 1}. This includes the class of all cofinite and finite sets. Before ending this section we would like to present the following theorem. It demonstrates the role of embeddings concerning the general problem of characterizing the injectively enumerable classes. T h e o r e m 5.1 A class ~ is injectively enumerable if and only if there is a recur- sive enumeration of 2 ~ which is embeddable in an injective enumeration of ~ ' := 2 ~ U { { 1 , 0 , 2 , . . . , 2 n } : n E w } . Here 2 ~ denotes {{2x : x e C} : C e ~ } . The proof of this theorem is trivial. The interesting point is however that the class ~ ' is injectively enumerable for any recursively enumerable class g~. This follows from many sufficient criteria, for instance Martin Kummer's Extension Lemma [K], Howard and Pour El's proof of the injective enumerability of the class of all r.e. sets [P] and Marchenkov's chain condition [M]. Injective enumerability:classes of cofinite sets 195 6 Laehlan's property (F) is equivalent to (G) After the following preliminary definition let us look at the property (F) (originally defined in [L, p. 215]): Definition 6.1 Let 6 be an enumeration of all finite sequences of natural numbers. We write 6(0,6 (0 for the i-th sequence, the n-th number of the i-th sequence, respectively. 6 should be such that the predicates x = 6~i),x = length(6 (i)) are decidable and given a finite sequence ~- = (no,... ,nk), or ~- = 0 an index i can be computed with 6(i) = ~-. For a finite sequence ~ and a number i let ~^i denote the sequence obtained by adding i to the end of or. Definition 6.2 An infinite r.e. class W has the property (F) if there is a partial recursive function r such that ,L, then r .[ and 6 (r C_ 6(r 1. if 6(i) C_ ~Sq) and r 2. if r ]., then D6~i) C W6c~<i),E ~ and We~e(~))=/WeU,)) for all m , n such that n -7' m < length(6 (i)) 3. if r is defined or 6(i) = 0 and D] is a finite set such that there is a set C E ~ with Dj C C different from all sets We~,c~)>, n < length(6 (i)) then r ,[ for all k such that 6 (k) = 6(i)~]. In his paper [L] Lachlan shows that this property is necessary for a class of r.e. sets to be r.e. without repetition and then proves that if ~ is an infinite r.e. class of finite sets which has the property (F), then ~ 4 is r.e. without repetition. Here we show Theorem 6.3 The properties (F) and (G) are equivalent. Proof of (G) ~ (F) If a class ~ satisfies (G), then (because (F) is necessary for injectively enumerable classes) the injectively enumerable subclass has property (F) and by the cardinality condition of the subclass given by (G) know that (F) also holds for W. Proof of(F) ~ (G) Given an r.e. class W for which (F) holds, witnessed by partial recursive function r we enumerate a set A of r.e. indices of sets of such that the enumerated class {Wi : i E A} witnesses that ~ satisfies (G). Step 0: let A be the empty set and r the empty sequence. Step s + 1: For all finite sets D o , . . . ,Ds and initial subsequences a r-- T, check whether r holds on the sequence c~^i in s steps. If so, wait for 1. r to hold on T^i or 2. the set Di to be enumerated in one of the sets Wz such that z entered A since T was extended from o. One of these two processes must terminate. If the first one does, then enumerate the given index in A and extend ~- by i. If the second search is successful, do nothing. End of construction. the we the ~, 196 s. Wehner By the properties of r the e n u m e r a t i o n given by A is injective. Suppose the set D i to have (at least) n different extensions in 4 . If at step so A contains n - 1 indices of extensions of Di and m is the largest index with Tm = i at stage so, then r has to terminate in t steps to extend the sequence ( T o , . . . ,~-m) for some step t. Now at step max{so, t} either one of the sets Wj such t h a t j entered A since r was extended from ( T o , . . . , Tin) contains D i or r has to hold on (T0,...,"rm)^i. In both cases A contains n indices of sets extending D . [] References [K] [KD] [L] [LM] [LC] [M] [P] [Yi] Kummer, M.: Numberings of Rl tO F, Computer Science Logic 1988. Lecture Notes in Computer Science 385, 166--186 (1989) Kummer, M.: Beitr~igezur Theorie der Numerierungen: Eindeutige Numerierungen. Dissertation, Universit~itKarlsruhe, Germany (1989) Lachlan, A.: On recursive enumeration without repetition. Z. Math. Logik Grundlag. Math 11, 209-220 (1965) Lachlan, A.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. (3) 16, 537-569 (1966) Lachlan, A.: On recursive enumeration without repetition: A correction. Z. Math. Logik Grundlag. Math. 13, 99-100 (1967) Marchenkov, S.S.: On minimal numerations of systems of recursively enumerable sets (abstract). Sov. Math. Dokl. 12, 843-845 (1971) Pour-El, M.B., Howard, W.A.: A structural criterion for recursive enumeration without repetition. Z. Math. Logik Grundlag. Math. 10, 105-114 (1964) Yi, X.: Manuscript