Coding by club-sequences

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


Given any subset A of ? there is a proper partial order which forces that the predicate x ? A and the predicate x ? ? {set minus} A can be expressed by ZFC-provably incompatible S formulas over the structure <H2, ?, N S1 >. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of H2 definable over <H2, ?, N S1 > by a provably antisymmetric S formula with two free variables. The proofs of these results involve a technique for manipulating the guessing properties of club-sequences defined on stationary subsets of ? at will in such a way that the S theory of <H2, ?, N S1 > with countable ordinals as parameters is forced to code a prescribed subset of ?. On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy.
Original languageEnglish
Pages (from-to)98-114
Number of pages17
JournalAnnals of Pure and Applied Logic
Issue number1-3
Publication statusPublished - 1 Oct 2006


  • Coding into Hω2
  • Optimal definitions over 〈Hω2,∈,NSω1〉
  • Club-guessing properties of club-sequences
  • Specifiable sequences of club-sequences

Cite this