SubmittedonSeptember3,2013totheNotreDameJournalofFormalLogic Volume??,Number??, 4 1 0 Guessing, Mind-changing, and the Second Ambiguous 2 Class n a J 9 Samuel Alexander ] O L Abstract Inhisdissertation, Wadge defined anotionof guessability on sub- . setsoftheBairespaceandgavetwocharacterizationsofguessablesets.Asetis h t guessableiffitisinthesecondambiguousclass(DDD 02),iffitiseventuallyannihi- a latedbyacertainremainder. Wesimplifythisremainderandgiveanewproof m ofthelatterequivalence. Wethenintroduceanotionofguessingwithanordinal [ limitonhowoftenonecanchangeone’smind. Weshowthatforeveryordinal a ,aguessablesetisannihilatedbya applicationsofthesimplifiedremainderif 1 andonlyifitisguessablewithfewerthana mindchanges. Weuseguessability v withfewerthana mindchangestogiveasemi-characterizationoftheHausdorff 4 9 difference hierarchy, and indicate how Wadge’s notion of guessability can be 8 generalized to higher-order guessability, providing characterizations of DDD 0a for 1 allsuccessorordinalsa >1. . 1 0 1 Introduction 4 1 Let NN be the set of sequences s:N→N and let N0 an ordinal, define Rmm (A,B)= Rmn (A,B)∩A∩Rmn (A,B)∩B . n\0,Sisguessablewhilechangingone’smind fewerthana +1manytimesifandonlyifatleastoneofSorSc isinthea thlevel ofthedifferencehierarchy. In Section 5, we generalize guessability, introducing the notion of m th-order guessability (1≤ m 0, G(s )≡imod2. 0 6= 1 6= i ThiscontradictsDefinition1.1sincelimn→¥ G((∪is i)↾n)oughttoconverge. The⇐directionrequiresalittlemachinery. Definition 2.4 If s ∈ N 0, then let G (s )=G (s −)wheres − isobtainedfroms byremovingthelastterm. S S (ii) Iftherearenoextensionsofs in[Sb (s )−1],andlength(s )=0,letGS(s )=0. (iii) If there are extensions of s in [Sb (s )−1] and they are all in S, define G (s )=1. S (iv) If there are extensions of s in [Sb (s )−1] and they are all in Sc, define G (s )=0. S Proposition2.8 IfS¥ =0/ thenGS guessesS. Proof Assume S¥ =0/. Let f ∈S. I will show GS(f ↾n)→1 as n→¥ . Since f 6∈[S¥ ],letibeasinLemma2.6. IclaimGS(f ↾ j)=1whenever j≥i. Fix j≥i. Wehaveb (f ↾ j)=b (f ↾i)bychoiceofi,and f ∈[Sb (f↾i)−1]=[Sb (f↾j)−1]. Since f ↾ jhasoneextension(namely f itself)inboth[Sb (f↾j)−1]andS,GS(f ↾ j)=1. Identicalreasoningshowsthatif f 6∈Sthenlimn→¥ GS(f ↾n)=0. Theorem2.9 S∈DDD 02 ifandonlyifS¥ =0/. Thatis,Theorem1.4istrue. Proof BycombiningPropositions2.3and2.8andTheorem1.2. 3 Guessingwithoutchangingone’sMindtoooften In this section our goal is to tease out additional information about DDD 0 from the 2 operationdefinedinDefinition2.1. Definition3.1 ForeachfunctionGwithdomainN0. The followingareequivalent. (i) Sisguessablewith0. If S∈Da , say S=Da ((Ah )h 0. IfS∈Da orSc∈Da thenSisguessablewith

Guessing, Mind-changing, and the Second Ambiguous Class

Similar Guessing, Mind-changing, and the Second Ambiguous Class