Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture

Transcription

1 Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture Peter Cholak Department of Mathematics Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.1/28

2 Acknowledgments Work is joint with Leo Harrington and Rod Downey. My research was/is partially supported by NSF Grants DMS , , and These slides are available at cholak/papers/nyc.pdf. Thanks for the invitation. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.2/28

3 Our Domain of Discourse Definition.A Âiff there is a map,φ, from the c.e. sets to the c.e. sets preserving inclusion,, (so Φ Aut(E)) such thatφ(a)=â. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.3/28

4 Our Domain of Discourse Definition.A Âiff there is a map,φ, from the c.e. sets to the c.e. sets preserving inclusion,, (so Φ Aut(E)) such thatφ(a)=â. Note thate can be replaced withe,e modulo the ideal of finite sets, as long asais not finite or cofinite. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.3/28

5 Slaman Woodin Conjecture Slaman Woodin Conjecture (1989). The set { i,j :W i W j )} isσ 1 1 complete. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.4/28

6 Slaman Woodin Conjecture Slaman Woodin Conjecture (1989). The set { i,j :W i W j )} isσ 1 1 complete. Theorem (Cholak, Downey, and Harrington (1995)). The Slaman Woodin Conjecture is true. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.4/28

7 A Stronger Result The Main Result. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.5/28

8 A Stronger Result The Main Result. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Hence Not all orbits are elementarily definable. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.5/28

9 A Stronger Result The Main Result. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Hence Not all orbits are elementarily definable. No arithmetic description of all orbits ofe Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.5/28

10 A Stronger Result The Main Result. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Hence Not all orbits are elementarily definable. No arithmetic description of all orbits ofe Scott rank ofe isω CK Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.5/28

11 A Stronger Result The Main Result. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Hence Not all orbits are elementarily definable. No arithmetic description of all orbits ofe Scott rank ofe isω CK (from the proof) For allα 4, there is a properly 0 α orbit. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.5/28

12 Why make such a conjecture? Certainly the set{ i,j :W i W j )} isσ 1 1. Why complete? Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.6/28

13 Why make such a conjecture? Theorem (Remmel/Folklore). There is a computable listing,b i, of all computable Boolean algebras such that the set{ i,j :B i B j } is Σ 1 1 complete. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.6/28

14 Why make such a conjecture? Theorem (Remmel/Folklore). There is a computable listing,b i, of all computable Boolean algebras such that the set{ i,j :B i B j } is Σ 1 1 complete. Definition.L (A) is{w A :W a c.e. set} under modulo the ideal of finite sets (F). Theorem (Lachlan (1968)). There is c.e. seth i such thatl (H i ) B i. Corollary. The set{ i,j :L (H i ) L (H j )} is Σ 1 1 complete. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.6/28

15 Why make such a conjecture? Theorem (Remmel/Folklore). There is a computable listing,b i, of all computable Boolean algebras such that the set{ i,j :B i B j } is Σ 1 1 complete. Definition.L (A) is{w A :W a c.e. set} under modulo the ideal of finite sets (F). Theorem (Lachlan (1968)). There is c.e. seth i such thatl (H i ) B i. Corollary. The set{ i,j :L (H i ) L (H j )} is Σ 1 1 complete. So replace L (H i ) L (H j ) with H i H j. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.6/28

16 D hhsimple Sets Definition (The sets disjoint froma). D(A)={B : W(B A W andw A= )} under inclusion. LetE D(A) bee modulod(a). Lemma. IfAis simple thene D(A) 0 3 L (A). A isd hhsimple iffe D(A) is a Boolean algebra. Except for the creative sets, until recently all known orbits were orbits ofd hhsimple sets. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.7/28

17 Complexity Restrictions Theorem. IfAisD hhsimple andaandâare in the same orbit thene D(A) 0 3 E D(Â). Theorem (using Maass 84). IfAisD hhsimple and simple (i.e., hhsimple) thena Âiff L (A) 0 3 L (Â). Hence the Slaman Woodin plan of attack fails. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.8/28

18 Related complexity restrictions Theorem. IfAandÂare automorphic thene D(A) ande D(Â) are 0 6 isomorphic. Hence must code aσ 1 1 complete set intod(a). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.9/28

19 Related complexity restrictions Theorem. IfAandÂare automorphic thene D(A) ande D(Â) are 0 6 isomorphic. Hence must code aσ 1 1 complete set intod(a). Theorem. IfAis simple thena ÂiffA 0 6 Â. Theorem. IfAandÂare both promptly simple thena ÂiffA 0 3 Â. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.9/28

20 Proving Main Theorem Main Theorem. There is a c.e. setasuch that the index set{i :W i A} isσ 1 1 complete. Theorem (Folklore). There is a listing,t i, of all computable infinite branching trees such that, for allα<ω CK 1, there is a tree,t α such that the set {i :T α T i } is α complete and there is a treet Σ 1 1 such that the set{i :T Σ 1 1 T i } isσ 1 1 complete. Idea. CodeT into the orbit ofa. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.10/28

21 The game plan 1. Coding: For eacht build ana T such that T T(A T ) via aα T 0 (2). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.11/28

22 The game plan 1. Coding: For eacht build ana T such that T T(A T ) via aα T 0 (2). 2. Coding is Preserved under automorphic images: If A T viaφthent(â) exists and T(Â) T viaα Φ, whereα Φ T Φ 0 (3). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.11/28

23 The game plan 1. Coding: For eacht build ana T such that T T(A T ) via aα T 0 (2). 2. Coding is Preserved under automorphic images: If A T viaφthent(â) exists and T(Â) T viaα Φ, whereα Φ T Φ 0 (3). 3. Sets coding isomorphic trees belong to the Same Orbit: IfT T viaαthena T A T via Φ α whereφ α T α 0 (2). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.11/28

24 The hidden parameters IfT(A) existed for everyathent(a) would be an invariant completely determining the orbit of A. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.12/28

25 The hidden parameters IfT(A) existed for everyathent(a) would be an invariant completely determining the orbit of A. Unfortunately infinitely additional parameters are needed and constructed witha T (Coding). The needed properties of these parameters but not the parameters themselves are preserved under automorphisms (Preserving). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.12/28

26 The hidden parameters IfT(A) existed for everyathent(a) would be an invariant completely determining the orbit of A. Unfortunately infinitely additional parameters are needed and constructed witha T (Coding). The needed properties of these parameters but not the parameters themselves are preserved under automorphisms (Preserving). Question. Can we make the additional parameters definable froma? Can we maket(a) into an invariant? Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.12/28

27 Coding The Starting Point Let R={R i :i ω} be a list of pairwise disjoint infinite computable sets such that for allw e, W e l 2e+1R l orw e l 2e+1R l = ω. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.13/28

28 Coding The Starting Point Let R={R i :i ω} be a list of pairwise disjoint infinite computable sets such that for allw e, W e l 2e+1R l orw e l 2e+1R l = ω. Theorem (Good News). R exists. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.13/28

29 Coding The Starting Point Let R={R i :i ω} be a list of pairwise disjoint infinite computable sets such that for allw e, W e l 2e+1R l orw e l 2e+1R l = ω. Theorem (Good News). R exists. Lemma (Bad News). There are no effective such lists. Ask ifw e \ i<2er i is infinite and build ther 2e and R 2e+1 accordingly (the odd sets are the error due to the needed tree construction). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.13/28

30 Coding A Ifiis even insider i we will construct a maximal subset,m i, ofr i. Ifiis odd thenm i =R i. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.14/28

31 Coding A Ifiis even insider i we will construct a maximal subset,m i, ofr i. Ifiis odd thenm i =R i. Using the Friedberg Splitting Theorem effectively split eachm i into finitely many Friedberg splits, H i,j. Ie.H i,j is hemi maximal insider i. Hence if W ցm i is infinite thenw ցh i,j is infinite. (We will assume there are just enough splits for our needs) Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.14/28

32 Coding A Ifiis even insider i we will construct a maximal subset,m i, ofr i. Ifiis odd thenm i =R i. Using the Friedberg Splitting Theorem effectively split eachm i into finitely many Friedberg splits, H i,j. Ie.H i,j is hemi maximal insider i. Hence if W ցm i is infinite thenw ցh i,j is infinite. (We will assume there are just enough splits for our needs) LetA= ih i,0. TheH i,0 s are the computable splits ofa. LetD 1 =A for notation ease. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.14/28

33 Coding the nodes oft Letf T be an onto map fromnto the nodes oft such that ifσ τ thenf 1 (σ)<f 1 (τ). We will codet by pairwise disjoint sets, {D j :j σ}. Iff(j)=σ thend j =D σ. D j = f(j) f(i)h 2i,(2i) j i H 2i+1,(2i+1) M i ( A jd j ) is infinite (and computable set) iff i is odd (som i =R i ). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.15/28

34 Coding R and theds These lists will be built such that if W e \ (A ) D e is infinite l<2er e l<e then and otherwise ( l 2e)[R l W e ] W e ( A l 2e+1 R l l 2e+1 D l ). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.16/28

35 Preserving Main Tool Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.17/28

36 Preserving Main Tool Theorem (The Conversion Theorem). IfAand are automorphic viaψ then they are automorphic viaφwhereφ L (A)=Ψ andφ E (A) is 0 3. Hence we will assume that we have a nice (computable in 0 ) listing of theψ(h i,0 )=Ĥ i,0. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.17/28

37 Preserving Maximal Supports Definition. X is supported bys ifs is a split of A,S X and(x A) S is a computably enumerable set). X is maximally supported bys ifx is supported bys and for allw, ifw is supported bys, thenw D X A. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.18/28

38 Preserving Maximal Supports Definition. X is supported bys ifs is a split of A,S X and(x A) S is a computably enumerable set). X is maximally supported bys ifx is supported bys and for allw, ifw is supported bys, thenw D X A. This is a definable property hence preserved by automorphisms. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.18/28

39 Preserving Maximal Supports Definition. X is supported bys ifs is a split of A,S X and(x A) S is a computably enumerable set). X is maximally supported bys ifx is supported bys and for allw, ifw is supported bys, thenw D X A. This is a definable property hence preserved by automorphisms. Lemma.H 2i,0 maximally supportsr 2l iffl=i. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.18/28

40 Preserving recovering the M 2i s Look (using 0 (3) andφ) for disjoint R i and disjoint M i s.t. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.19/28

41 Preserving recovering the M 2i s Look (using 0 (3) andφ) for disjoint R i and disjoint M i s.t. Ĥ 2i,0 maximally supports R 2i. for allŵ there is anisuch that either Ŵ l i R l,ŵ l i R l = ω, orŵ Â is r.e. outside l i R l. M 2i is maximal in R 2i. For each 2i,Ĥ 2i,0 is hemi maximal in M 2i. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.19/28

42 Preserving recovering the tree Let D i =Φ(D i ). These code the nodes. D j is below (in the tree T) D i iff for alll, if D i M 2l is infinite then D j M 2l is infinite. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.20/28

43 Same Orbit Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.21/28

44 Same Orbit Recall: Our sets will be built such that if W e \ (A R e ) D e is infinite l<2e l<2e then and otherwise ( l 2e)[R l W e ] W e ( A l 2e+1 R l l 2e+1 D l ). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.21/28

45 Same Orbit where defineφ? EitherW e ( A l 2e+1D l ), Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.22/28

46 Same Orbit where defineφ? ( EitherW e A l 2e+1D l ), W e (A l 2e+1D l l 2e+1(R j ) l FD l ), here eitherr j l FD l is maximal inr j (j is even) orr j l FD l is a computable subset ofa (j is odd). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.22/28

47 Same Orbit where defineφ? ( EitherW e A l 2e+1D l ), W e (A l 2e+1D l l 2e+1(R j ) l FD l ), here eitherr j l FD l is maximal inr j (j is even) orr j l FD l is a computable subset ofa (j is odd). or W e i<2er i l 2e+1D l = ω. In this case there is a computablersuch thatr W e =ωand mapw =R (W R) toφ(r) Φ(W R). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.22/28

48 Same Orbit Enough To defineφtakingatoâ, it is enough to determine where wheree (D i ) goes, as long as the computable subsets go to computable sets, ifiis even wherer i goes (R i must go to computable set), and ifiis odd wheree (R i l F D l ) goes (R i l F D l must go to a computable subset of Â). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.23/28

49 Same Orbit extendible algebras The set ofr j such thatd i R j andd i R j is infinite forms an extendible algebra of computable sets,b i. Furthermore there is a natural 0 3 isomorphism,θ, between an extendible algebra,b and an extendible algebra, B (if these algebras have the same size). This isomorphism is extendible. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.24/28

50 Same Orbit Main Tool Theorem. LetBbe a extendible algebra of computable sets and similarly for B. Assume the two are extendibly isomorphic viaθ. Then there is aφsuch thatφis a 0 3 isomorphism between E (D) ande ( D),Θmaps computable subsets to computable subsets, and, for allr B, (Θ(R) D) Φ(R D) is computable (and dually). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.25/28

51 Same Orbit Φ(D i ) and more AssumeT and T are isomorphic viaα. Assume α(i)=f 1 T (α(f T(i))).α( 1)= 1. LetΘ i be the natural extendible map fromb i to B α(i) and letφextend eachθ i to mape (D i ) to E ( D α(i) ) via the last theorem. Letibe odd.r i l FD l is infinite, computable, and not in anyds. There is a natural effective map fromr i l FD l to R i l F D l. LetΦextend this map for all oddi. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.26/28

52 Same Orbit Φ(R 2i ) Given anr 2i there are finitely manyd j such that R 2i is the extendible algebrab j. For exactly these samej, R 2α(i) is in B α(j). Hence ( R 2α(i) D α(j) ) (Φ(Θ 1 j ( R 2α(i) ) D j ) is computable. MapR i to ( R 2α(i) ) D α(j) j F j F (Φ(Θ 1 j ( R 2α(i) ) D j ). Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.27/28

53 Thanks For the invitation. For listening and your interest. Progress on the c.e. sets: Improving and Proving the Slaman Woodin conjecture p.28/28