DIRECT AND LOCAL DEFINITIONS OF THE TURING JUMP

RICHARD A. SHORE
2007 Journal of Mathematical Logic  
We show that there are 5 formulas in the language of the Turing degrees, D, with ,_ and^, that de...ne the relations x 00 y 00 , x 00 = y 00 and so x 2 L 2 (y) = fx yjx 00 = y 00 g in any jump ideal containing 0 (!) . There are also 6 & 6 and 8 formulas that de...ne the relations w = x 00 and w = x 0 , respectively, in any such ideal I. In the language with just the quanti...er complexity of each of these de...nitions increases by one. For a lower bound on de...nability, we show that no 2 or 2
more » ... ormula in the language with just de...nes L 2 or L 2 (y). Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of I is ...xed on every degree above 0 00 and every relation on I which is invariant under the double jump or under join with 0 00 is de...nable over I if and only if it is de...nable in second order arithmetic with set quanti...cation ranging over sets whose degrees are in I.
doi:10.1142/s0219061307000676 fatcat:ymhxmsnwjrbidpz3q3krj3emvi