Index sets of c.e. superset classes of r-maximal sets

LetA²R^;Max. We will consider here the index sets of FINA =^fWe: We^\A is nite^g and of COF_A =^fWe :AWe = IN^g.

The index sets FINA are "₀₃-complete for any r-maximal, not maximal set A. This follows at once from the fact shown by Maass in Ma85]. He proved that FINA is

"03-complete for any A^2MSS. But R^;MaxⁿMax is a subclass of ^MSS. Thus we get the equivalence

A²R^;Max^{^} A²SIL2 ^,A maximal:

For the class COFA the situation is more complicated, i.e. is not still cleared for the r-maximal sets in general.

Easy to see is that for A ² R ^;Maxatm COFA T ⁰⁰. Since for A and B with AmB ^_A = B

WeA = IN ^,WeB = IN:

If B is maximal (then obviously COFB T ⁰⁰) we get COFA T ⁰⁰. But also some atomless,r-maximal sets have this property.

Lemma 5.6

Let A^2Tr. Then COFA²₀₃.

Proof.

Given A ^{2 T}r let B ^{2 L}(A) be a basis element for ^L;(A), i.e. for every C ^2L;(A) there are sets C1C2 ^2L(A) with C1 B, C2^\B = A s.th. C = C1C2. Then

We^\ A⁶= IN ^,(⁹f)(⁹k)(Wf B^{^}Wk^\B A^{^}We =Wf Wk): ² For the r-maximal set A constructed in Theorem 2.4 we also have COFA ² 3. It holds

AWe= IN i (⁹n)(⁸nn⁰)(Fn A^_FnⁿWe⁶=):

"FnⁿWe" is 1, since (Fn)n0 is strongly c.e.

Easy to see is also that forA²R^;Max for allB and C from A]=ms COF_B = COF_C. We have

B ms C⁾(BWe = IN ^,CWe= IN):

The r-maximal sets A with COFA ² 3 have a property which seems to be useful for the characterization of ^L(A), what in the following we discuss. We consider a weaken of the notion of tower from subpoint 2.

Call a sequence (Hn)n0 (not necessarily c.e.) of c.e. setsweak tower if for alln 0Hn Hn+1 and Hn⁶= IN.

Obviously if for a coinnite c.e. setA there is a weak tower (Hn)n0 with A H0 such that (2.1) is satised thenA is r-maximal (not necessarily atomless). Interesting is the case when forA there is a weak tower which is c.e., i.e. (Hn)n0 { the weak tower { is an c.e. sequence.

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Lemma 5.7

Let A be a r-maximal set. Then A has a weak c.e. tower (Hn)n0 with A H^{0 satisfying} (2.1) i COFA T ⁰⁰.

Proof.

⁾ Suppose A has an c.e. weak tower (Hn)n0 with A H0 and (2.1) is satised. Then we have

AWe⁶= IN i (⁹n)(We Hn) e0:

Thus "AWe⁶= IN" belongs to "03 and thus COFA to ₀₃.

) Suppose COF_A ²₀₃. Let be a computable function with e²COFA i (⁸i)(e(i) < !) e0:

LetUei be the set

fWes :e(is + 1)⁶=e(is) s0^g:

We see that for e =² COFA there is an i with e(i) = !, hence Uei = We and for e²COF_A then for alle and i Ueiis nite. Let forx0 Vx be equal to ^S

heiⁱxUeiA.

Then (Vx)x0 is an c.e. sequence and is a tower for A with A V0 satisfying (2.1).² Thus if for A ² R ^;MaxCOFA then A has a c.e. weak tower. But the existence of a weak c.e. tower for A implies the existence of a computable function f such that (Wf(e))e0 "bounds" every coinnite c.e. superset of A by condition (2.1), what together with the e ectivity presented isomorphism between major subset intervals, shown by Maass and Stob, would gives a good tool for charactering ^L;(A).

Sorely not every r-maximal set has a c.e. weak tower what from the next theorem in connection with Lemma 5.7 follows.

Theorem 5.8 (Nies, Lempp, Solomon)

There is an r-maximal set Awith COFA{

"3-complete.

Proof.

LetQ be a "3-predicate. We construct anr-maximal set A and a c.e. sequence of sets (Ce)e0 such that

Q(e)⁽⁾ACe = IN:

In the rst part of the proof we describe the construction of the setA. For doing this at rst we give the basic idea of this construction. After itA will be constructed. This will be an innitely many simultaneous application of the slightly modied basic method.

This approach seems to be good for the understanding of the general construction. In the second part of the proof the description of (Ce)e0 is given.

Let (R0eR1e)e0 be a duple c.e. sequence with { R0eR1e =IN or R0e R1e is nite, { R_0e^\R_1e =, e0,

36

{ (⁸e)(We^;computable ⁾(⁹f)(We=R0f)).

(Basic construction)

. In the following we construct a coinnite setC and innitely many pairwise disjoint subsets Mi of IN, i0, with ^SfMi :i0^g=IN and for some e all sets Mi,i < e, are nite, Me^;C is innite and for all j > e Mj^;C are empty.

(In the general construction everyMi becomes decomposed in the same way into sets Mij, j > i, every set Mij into sets Mijk,k > j, and so on.)

We construct C stepwise and together with C a function T such that Ts gives a nite list of elements of Mis at steps. The number of these lists also is nite. Thus

f(ij) : Ts(ij)^#g; nite Ts(ij)^{#^}j⁰ j ⁾Ts(ij⁰)^#:

Hence^fj : Ts(ij)^#gis a nite initial part of IN. The greatest j such that Ts(ij)^#(if Ts(i0)^#) is of particulary interest. We denote it with xis.

Step 0. Dene C0=, x00= 0, T0(0x00) = 0. (For all other pairs (ij) T0(ij)^".) Step s + 1. We have Cs, numbers x0sx1s:::xss and Ts injective with Ts(ij) ^, is^{^}j xis.

Look if there is an es such that 0Ts(exes)] R0esR1es: (5.1)

{ If not let all unchanced and dene xs+1s+1 = 0 and Ts+1(s + 10)-equal to a fresh element (i.e. equal to a number which is not in Cs+1 and is not one of the numbersTs(ij), (ij)²IN, for which Ts(ij)^#.

{ If yes let e be the smallest such number and do the following:

(i) Put all numbers z max^fTs(ij) : Ts(ij)^#g which are not of the form:

Ts(ik), ie, k xis to Cs+1.

(ii) If Ts(exes) ² R0es than put additionally all numbers Ts(ek) ² R1es (k <

xes) to Cs+1. Now dene

xis+1 = xis :i < e xis+1 = 0 :e < is + 1 xis+1^" for i > s + 1

xes+1 = xes+ 1 if Ts(exes)²R_1es

= ^jfk : Ts(ek)²R0es kxes+1^gj+ 1 if Ts(exes)²R0es: Further

Ts+1(ik) = Ts(ik) i < e k xis

Ts+1(ei) = Ts(ei) i < xes+1^;1

= Ts(exes) i = xes+1 ^;1

Ts+1(ixis+1) ^; a fresh element for every i with eis + 1:

37

(Of course di erent pairs become assigned di erent fresh numbers thus Ts+1 is injec-tive.)

Result. Let C =^SfCs :s0^g. We see that C is a c.e. set.

1. Let f = (z)(R0zR_iz = IN). Then for e < f (1) can happen only nitely often.

Lets0 be such that for all ss0 (1) does not hold for an e < f.

Letxi=df limsxis,i < f. Then for i < f T(ij) =^df limsTs(ij) exists i j xi. But this are only nitely many pairs. For s > s0 Ts(fl), l < xfs cannot come to Cs+1

by (i), but only by (ii). Hence Ts(fl) ⁶=Ts+1(fl) can be at most one time. Further by denition of f (1) holds innitely often for e = f and xfs^# for all s f. Hence limsxfs =¹. Let T(fl) =df limsTs(fl). Then T(fl) =²C for all l0. HenceC is coinnite. For e > f xes = 0 for innitely many s and limsTs(exes)^".

2. From construction case (ii) we get C R0f or C R1f:

3. We see that the numbers Ts(ixis) for i s (xis^#) are the onliest among Ts(ij) withTs(ij)^#which can chance their places, i.e. that can beTs(ixis) =T(il), l < xis

for somet > s. For all other numbers Ts(ij), j < xis either they are putted intoC or for allt > s Ts(ij) = Tt(ij).

(The places of the Ts(ixis)'s are still not determined.)

4. The setsMi about which was spoken in the beginning of the construction are equal to

si^fTs(ik) : 0kxis^g:

(The sets Mi \measure" the growth of the set s R_0isR_1is.)

More interesting are other sets which we denote with Xj. LetXj be equal to

s0

i0

fTs(ij) : Ts(ij)^{#^}Ts(ij) ⁶=Ts(ixis)^g:

(Xj)j0 is a c.e. sequence of disjoint sets withXj ^\ C-nite, X0 ^\ C ⁶= (Xi^\ C ⁶= for every i 0, but we do not need it) and C ^SfXi : i 0^g. (We have only , since the numbersT(ixi),i < f are not in the union.)

(General construction, construction of

A

)

. Let ; be the set

f² IN^<! : ⁶=^hi = (01:::n^;1) 0 < 1< ::: < n^;1^g:

Further let 01::: be a computable sequence of the elements of ; such that every element of ; appears innitely often in this sequence.

We construct a function Ts, set Ans, n 0 with A0n A1n ::: (A0s has the same meaning asCs from the basic construction andAnsfor the similar construction on level n) and numbers xs, ²;. (In every steps only nitely many of them are dened.) Step 0. Dene An0 = for all n 0 and all other objects, i.e. x0 and T0(i) are undened.

38

Step s + 1. We have Ans, n0, xs for nitely many's from ; and xs^#)Ts(i)^# for ixs:

(The converse, i.e. for someixs Ts(i)^#)xs^# in general is not true.) Further we assume that Ts( ) is injective in .

Takes. (In the following we write only for s.) Look ifx s^#. { If not, look if there is a fresh element on

fTs(^;1)Ts(^;2):::Ts(^;l)^g;A^j_s^j;1 (5.2)

where l is the greatest number for which Ts(^;l)^# and lesser thanx ^;s (if x ^;s^#)⁶⁾.

\Fresh" means not equal toTs(^;ij) for no i and j.

If there is a fresh element let x s+1 be equal to max^fl : Ts(l)^#g+ 1

and Ts+1(x s+1) be this fresh element. If there is no fresh element go to the next step.

{ Ifx s^# then look if

0Ts(x s)] R0esR1es (5.3)

where = ^;e. If not go to the next step. If yes then (iii) Take all z with

z max^fTs(^;ij) : Ts(^;ij)^#g

which are not of the formTs(^;ik), i e, ^;i ²; to Ans+1 for all n^j;j. (iv) If Ts(x s) ² R_0es then take additionally all Ts(k) ² Rles (k ⁶= 0) to Ans+1

(n^j;j). (k ⁶= 0 above will be ensure that A will be coinnite.) Now dene

x s+1=x s + 1 if Ts(x s)²R_les and there is a fresh element in

fTs(^;i) : 0 < i < x ^;s^g for Ts+1(x s+1):

If by (iii) a number Ts(i) comes to Ans+1 then Ts+1(i)^" (except that = and l becomesx s+1. We see that if this holds forTs(i) then also for all Ts(j) with i < j for which Ts(j)^". Thus after every step for every ² ; Ts+1() is a initial part of IN.

Result. LetAⁿ=_s^S

0Ans and A =_n^S

0Aⁿ.

1. We haveA⁰ A¹ ::: and all sets and A are c.e. By the construction (Ans)ns0

is a strong c.e. sequence of nite sets.

2. Let e0 < e1 < ::: be all indices in order of magnitude such that R_0iR_1i =IN i i is one of the ej's.

6)^; means the immediate predecessor of . If ^;=^hiinstead of the set (5.2) take the set INⁿA⁰, and without the restriction for l.

39

Then for = (e0:::eke) with e < ek+1 the requirement (5.3) can happen only nitely

The generalization of the setsXj from the basic construction are the sets X_k⁽ⁿ⁾, n1, k 0. Let Xk⁽ⁿ⁾ be the sets

The sets X_k⁽ⁿ⁾ have following properties:

1 X0^(n)\ A⁶=.

The sets in the triangles and X0⁽³⁾ have no common elements in A. (But X0^(3)\ A⁶=.) Fixe. In the following we write Cs andUisforCes andUeis respectively. In steps+1 we have a sequence of numbers 0 < a0s < a1s < ::: < ass with ^jUis^j < ais, i s, ais ais+1 and ^jUi^j<^1,limsais. Let ai be the limit if it exists. LetCs be

i+1=0:::s

kj

fXk^(j):ais< k + j < ai+1s^g (a^;1 =df 0) (5.4)

(C0 =df ). C =^SfCs: s0^g.

If ais^"1, but ai^;1^# then ^SfXi^(a+1);1 :k 0^g C, see (5.4). Hence A C. If for all i ai^# then (_i^S

0Xi^(a)0 ⁿA)^\C = and _i^S

0Xi^(a)0 ⁿA is innite. Hence ACe ⁶= IN. ²

Question 5.9

Exists for every r-maximal set A an r-maximal set B with a c.e. weak tower such that ^L(A)=^L(B) (or even more A=^E B)^?

6

^r

-Maximal sets, co-Monotone and co-1-1 sets

In MadRob82] there were investigated notions which are closely related with the class R^;Max more precisely which form a subclass of R^;Max. These notions are:

Denition 6.1

LetX IN be innite. X is called monotonic (1{1) if (⁸f rec. fct)^h(⁹n)(⁸mm⁰n)(mm⁰²X mm⁰

)f(m)f(m⁰))^_f is constant on X (mod = )ⁱ: (6.1)

(⁸f rec. fct)(⁹n)(⁸mm⁰n)(mm⁰²X m⁶=m⁰

)f(m)⁶=f(n))^_f is constant on X (mod = )]: (6.2)

We are interested here in coinnite c.e. sets having monotonic or 1{1 complements.

The following facts hold, see MadRob82]:

1) A coinnite c.e. set has a monotonic complement i it has a 1{1 complement.

2) Every maximal set has a monotonic (and thus by 1)) also a 1{1 complement (Owings).

3) Every innite set which is monotonic or 1{1 is dense immune. Thus together with 1) and 2) we get that theT-degrees of the monotonic sets as also of the 1{1 sets is equal to IH¹.

41

6.1 The position of the co-monotonic sets inside ^R

^;

^Max

Denote with Co-Mon (Co{1{1) the class of coinnite c.e. sets with monotonic (1{1) complements. From 1) we get Co{Mon=Co{1{1, from 2) Max Co{Mon and from 3) Co{Mon ^{D S} (the class of dense simple sets).

Lemma 6.2

Every A fromCo{Mon belongs to R^;Max.

Proof.

LetX be an innite set not r-cohesive. Then there is a computable set R such that R^\X and R^\X are both innite. Let f(x) = 0 for x ² R and f(x) = 1 for x ² R. Then f is neither monotonic (mod = ) on X, nor 1{1 (mod = ) on X. Thus for a coinnite c.e. set A if A is not r-cohesive then A is not monotonic and not 1{1.

2

In the following Lemma 6.3 we need a fact also shown in MadRob82]:

4) Let W be an c.e. set and (Sn)n0 be an c.e. sequence of disjoint sets. Then a) There is a disjoint c.e. sequence (Tn)n0 with_n^S

0Tn=_n^S

0SnandTn^\ W ⁶= for all n or

b) there is a computable function g such that for almost all n (⁸x)(x²Sn^\ W ⁾xg(n)):

Lemma 6.3 (Madan, Robinson)

If Y ²Co{Mon andX mY then X ² Co{Mon.

Proof.

Let f be a computable function not constant (mod = ) on Y and Sn = f^;1(^fn^g), n 0. Since Y is monotonic, ^jSn^\ Y^j 1 for almost all n and Sn^\ Y is nite for alln. If for an n0 Sn0^\Y would be innite, since f is not constant (mod = ) on Y , f(x) = 0 for x ² Sn0 and f(x) = 1 for all other x would contradict that Y is monotonic.

Letn0 be such that forn n0 ^jSn^\ Y^j1. Then we have

Y

n0Sn and Y is r-cohesive, by Lemma 6.2:

Thus by fact 4) there is a computable functiong s.th.

(⁸n)(⁸x)(x²Sn^\ Y ⁾xg(n)):

Thus for all nn0

jSn^\ Y^j1^{^}x²Sn^\ Y ⁾xg(n):

(6.3)

Letxn be the last enumerated element from Sn with xg(n), not earlier enumerated into Y , for n n0. Let be the set of all these xn's. Then is c.e. and Y . Thus X and f is 1{1 on , since f(xn) =n. Hence f is 1{1 (mod = ) on X.

Iff is constant (mod = ) on Y, i.e. Y f^;1(^fk^g) for somek then also X f^;1(^fk^g), since f^;1(^fk^g) is c.e. Both together give that X is an 1{1 set. ²

42

Corollary 6.4

If A^2MSMax then A² Co{Mon.

Proof.

Since every maximal set is co-monotonic, see fact 2), from Lemma 6.3 we get

it also for its major subsets. ²

Remark.

From Lemma 6.3 we can conclude that all elements in an equivalence class resp. to ms are co-monotonic or none is it.

Suppose X ² Co{Mon and Y ² X]=_ms. Then Y ms X Y , hence Y X, what implies that Y is monotonic or Y m X Y . But X Y X, hence is monotonic and thus Y ²Co{Mon, by Lemma 6.3.

*

The next question concerns the relationship between Co{Mon and R ^;Max: Holds Co{Mon =R^;Maxatm or not?We get an answer by comparing both index sets.

Lemma 6.5

^fe : We ²Co{Mon^g is₀₄-complete.

Proof.

Since Max Co{Mon R ^; Max and obviously R ^; Max fs^HS, by using the fact in 5.1 we need only to show that "We ² Co{Mon" has a ₀₄-denition.

Considering (6.1) we have

We^{;1^8}f('f ^;total⁾⁹n⁸mn⁰(mm⁰²= We m < m⁰⁾'f(m) 'f(m⁰))

_9a⁹n⁸m > n(m =²We⁾'f(m) = a)]:

This is ₀₃^{^8}₀₂^!98(^8!9)^_989] ₀₄. ²

Corollary 6.6

R^;Maxatm⁶=Co{Mon.

Proof.

The index set of R ^;Maxatm is "₀₅-complete, see Theorem 5.3 and that of Co{Mon ₀₄-complete, see Lemma 6.5. Hence both classes cannot coincide. ²

Corollary 6.7

There are atomless, r-maximal sets which are co-monotonic.

Proof.

From Lemma 6.2 and Corollary 6.6. ²

Fact 3) and Lemma 6.2 together give the inclusion Co{Mon R^;Max^{\D S}. This inclusion was more intensively investigated in MadRob82] as the following theorem shows:

Theorem 6.8 (Madan, Robinson)

There is an r-maximal and dense simple set W, which is not co-monotonic.

Proof.

The construction of W will be a generalization of that in Theorem 2.4. Now we do not x at the beginning a nitely strong c.e. sequence for the whole construction as (Ei)i0 in Theorem 2.2 and in the steps we construct Fis with Fis+1 Fis Ei, but here it can happen that for some s Fis+1 is dened new in such a way that Fis+1^\Fis =. Since the elements of W are included into Fis, i 0, if Fis consists

43

only of su$ciently large elements, W will be dense immune. The constuction method from Theorem 2.2 and this "Fis-chance principle" together give that W is atomless, r-maximal and dense simple.

Since further the sets Fis, is0 are either included one in the other or are disjoint, the sequences are strongly nite and the sets Fis have at least two elements, it can be dened a recursicve function decreasing inside every Fis. This negates that W is monotonic.

Letrs(n) be the function:

max^f'es(n) : For all e < n for which (⁸xn)('es(n)^#) is satised^g 0 If such e does not exist:

Further let (An)n0 be a computable decompositon ofIN into innite, pairwise disjoint computable sets with _n^S

0An =IN. At every step s we dene an (increasing) function ys(n) with the meaning: Fns Ay^s(n).

We say that n requires attention at step s + 1 bye if

e < n^{^}Fns⁶ Wes+1^{^j}Fns^j2^jFns^\Wes+1^j:

Let stR(xes) be the special e + 1 state of x at step s + 1 dened by stR(xes)(j) = 1 if Fxs Wes+1

= 0 otherwise j e:

For two nite, nonempty sets XY IN we write X < Y , if ⁸x ² X ⁸y ² Y (x < y).

Construction.

Step 0:

{ Let y0(n) = n for all n0.

{ F0(0) { the set of the rst two elements of Ay0(0) in order of magnitude.

F0(n + 1) the rst 2ⁿ⁺² elements of Ay0(n+1) in order of magnitude such that F0(n) < F0(n + 1).

{ g will be dened on _n^S

0F0(n) in such a way that for

F0(n) =^fa1< a2 < ::: < a2n+1^g g(ai) = 2^n+1;i, i = 1:::2ⁿ⁺¹. { Let T0(n) be the increasing enumeration of _n^S

0F0(n) and W0 =.

Step s ^{+ 1 :}

If there is no n for which Ts(n) < rs+1(n) do nothing. Otherwise let n0 = (n)(Ts(n) < rs+1(n)).

Now we do the following (in order (i) to (iv)):

(i) Suppose Ts(n0) ²Fs(m0) Ay^s(m0) for somem0. For every m m0 dene new sets F^bs(m) inside Ay^s(m) such that

44

{ x ²F^bs(m0)⁾rs+1(n0)< x

;j

Fbs(m)^j= 2^m+1 (6.4)

{ F^bs(m) <F^bs(m + 1)

{ F^bs(m) consists of elements which are "new", i.e. are not in dom(g), not in any Wes, not in Ws, not inFs(m).

(ii) If (⁹n < m0)(⁹e < n) (n gets attention by e at step s+1) then take the smallest such n (let be this n1) and for this the smalleste (let be this e1) and dene

Fbs(k) = Fs(k)^\We1s : k = n1

= Fs(k) : 0 k < m0 k ⁶=n1:

(iii) If (⁹m)(⁹n)(⁹e)e m < n < m0 7)^{^}stR(mes) < stR(nes)] (where stR is dened by means ofF^bs(k) instead of Fs(k)) then m1 be the smallest suchm and n2 be the smallest n for m1. Dene

ys+1(m1 +k) = ys(n2+k) k 0 ys+1(k) = ys(k) : k < m¹ Fs+1(k) = F^bs(k) : k < m1

Fs+1(m1 +k) ^; the rst 2^m1+k+1 elements (6.5)

of F^bs(n2+k) : if ^jF^bs(n2+k)^j> 2^m1+k+1

= F^bs(n2+k) : otherwise k 0:

(6.6)

(iv) { Put into Ws+1 allxs with x =²_n=1^1S Fs+1(n) and all x from Ws. { Let Ts+1 be an increasing enumeration of _n^S

0Fs+1(n).

{ Deneg(x) = 0 for x²Ws+1 for whichg was not still dened and if Fs+1(n) if not still dened dene g similar as in step 0.

Result.

At rst we see that step s + 1 innitely often happens (i.e. there is an n withTs(n) < rs+1(n)). After every step the function Tsis computable. Hence for some s⁰s and some n Ts⁰(n) < rs⁰+1(n) and thus the construction (i) to (iv) will be taken place.

In the construction step s + 1 we start with ys, Fs, gs, Ts,Ws and after (i) we haveys, F^bs(m), mm0,gs,Ts, Ws

(ii) we haveys, F for all m, g^b s, Ts, Ws

7)The requirement "n < m⁰" is not necessary, since for n m⁰ st^R(n e s) = 0 by (i) and (ii) and thus st^R(m e s) < st^R(n e s) cannot hold.

45

(iii) we haveys+1, Fs+1, gs, Ts, Ws

(iv) we haveys+1, Fs+1, gs+1,Ts+1,Ws+1.

1. (⁸n)(⁹s)(⁸s⁰> s)(ys⁰(n) = ys(n)^{^}Fs⁰(n) = Fs(n)).

(By induction). Givenn let s0 be such that for k < n and ss0 ys(k) = ys0(k) and Fs(k) = Fs0(k).

Lets1 s0 be such that for all ss1 rs(k) = rs+1(k) for all k < 2^n+2;1. After step s1 n maybe some m m0 in (i) on behalf of some k < 2^n+2;1. But for everyk at most once, since by (6.4) and (6.6) ^j_i^S

nFs(i)^j< 2^n+2;1 for all s.

Hence there is a step s2 s1 such that for s s2 in step s + 1 in part (i) always n < m0.

Further we see that the parts (ii) and (iii) implyTs(n) < Ts+1(n), hence do not disturb that (i) is not satised for k < 2^n+1;1.

Part (iii) can happen only by a greater n-state of which there are only nitely many.

Thus ys(n) = ys3(n) for some s3 s2 and all s s3. But (Fs(n))ss3 is decreasing.

Thus for somes4 s3 Fs(n) = Fs4(n) for all ss4. LetF(n) = limsFs(n) and y(n) = limsys(n), n0.

2. (⁸n)(⁸s)(^jFs(n)^j2).

Given n let s0 be such that F(k) = Fs0(k) and y(k) = ys0(k) for k n. Thus F(n) = Ay(k). For s s0 let h(s) be such that Fs(h(s)) Ay(n). Then h(0) = y(n) and h(s0) =n and h is decreasing.

(This means Ftⁱ(h(ti)) Ay(n), i = 01:::k).

Let s1 < s0 be the greatest number such that (i) holds for somem h(s1) in step s1. (Ifs1 does not exist, let it be 0.)

Then at the end of step s1 ^jFs1(h(s1))^j= 2^h(s1)+1, by (1) and for alls with s1 < ss0

we haveF(n) Fs(h(s)) Fs1(h(s1)).

The setsFs(h(s)) can get attention by a number e at most one time. Then also e < h(s) must be and Fs(h(s)) becomes divided at mostly by 12.

Let s⁰ be the greatest number with s1 s⁰ < s0 such that h(s⁰) < h(s^0;1) and (3) holds for Fs⁰(h(s⁰)). Is such s⁰ does not exists let s⁰ =s1. Then ^jFs⁰(h(s⁰))^j = 2^h(s0)+1 and for all s⁰⁰ > s⁰ only (ii) holds for Fs⁰⁰(h(s⁰⁰)) by some e < h(s⁰⁰) h(s⁰) (and at most once). Thus ^jFs0(h(s⁰))^j 1

2^n(s0) 2^h(s0)+1 = 2. hence^jF(n)^j2.

3. W ist r-maximal.

SupposeR is computable and We0, We1 are R and R respectively. By the maximaliza-tion of the e-states for n with e < n either for almost all n > e and all e F(n) We

or ^jWe^\F(n)^j< 12 ^jF(n)^j.

If for both We0 and We1 the second case hold then ^jWe0 ^\Fn^j< 12^jFn^jand

jWe1 ^\Fn^j < 12^jFn^j for almost all n. But this gives (We0 We1)^\ A ⁶= A. Thus for one set the rst case must hold, i.e. F(n) We0 for almost all n. Hence A We0. Similar if F(n) We1 for almost all n. This gives that A is r-cohesive. ²

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Im Dokument r-Maximal Sets (Seite 35-47)