Defining Jump Classes in the Degrees below 0'1
نویسندگان
چکیده
We prove that, for each degree c r.e. in and above O'3', the class of degrees x < 0' with x'3) = c is definable without parameters in 3{< 0'), the degrees below 0'. Indeed the same definitions work below any r.e. degree r in place of 0'. Thus for each r.e. degree r, Th(^(< r)) uniquely determines
منابع مشابه
On the jumps of the degrees below an r.e. degree∗
A perhaps plausible conjecture about distinguishing among some r.e. degrees based on the jumps of degrees below them would prove the rigidity of the r.e. degrees. We show that the conjecture is false by exhibiting, for every c r.e. in and above 0′, distinct r.e. degrees a and b, each with jump c, such that the classes of the jumps of degrees below a and below b are the same.
متن کاملBenign cost functions and lowness properties
We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost function, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below inc...
متن کاملAntibasis theorems for Π1 classes and the jump hierarchy
We prove two antibasis theorems for Π1 classes. The first is a jump inversion theorem for Π1 classes with respect to the global structure of the Turing degrees. For any P ⊆ 2, define S(P ), the degree spectrum of P , to be the set of all Turing degrees a such that there exists A ∈ P of degree a. For any degree a ≥ 0, let Jump(a) = {b : b = a}. We prove that, for any a ≥ 0 and any Π1 class P , i...
متن کاملProperties of the jump classes
On the Turing degree structure we consider an ordering relation, which is that inherited by the Turing reducibility on sets of natural numbers, and also the jump function, which arises naturally as the function on degrees induced by the operator which takes each set to its halting problem. In order to understand the structure it seems a basic task to understand the relationship between these tw...
متن کاملThe Jump Operator on the Ω-enumeration Degrees
The jump operator on the ω-enumeration degrees is introduced in [11]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structureDω ′ of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of Dω ′ and of the enumeration degrees are isomorphic. ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010