The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

Reverse Mathematics and Algebraic Field Extensions

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section 2, we show that $\mathsf{WKL}_0$ is equivalent to the ability to extend $F$-automorphisms of field extensions to automorphisms of $\bar{F}$, the algebraic closure of $F$. Section 3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section 4, and the Galois correspondence theorems for infinite field extensions are treated in section 5.Comment: 25 page

Fields in Mathematical Logic

Chapter 1, Computable fields. We define computable fields, using the natural numbers NN. We show that if K is a computable field, then its algebraic closure K' is a computable field. We show that if K is a computable field such that the equality relation is computably enumerable, then its algebraic closure K' is a computable field such that the equality relation is computably enumerable. We show that if K is a computable field such that the equality relation is computable, then its algebraic closure K' is a computable field such that the equality relation is computable. With this definition of a computable field, we show that there is however a computable field with a non-computable property. Chapter 2, Model theory of fields. We use first order logic to look at the (+,.,0,1)-structures called fields. Let TF be the theory of fields. The complete extensions of TF gives the best descriptions of fields. Let Th K be the complete theory of a field K. We consider the isomorphism classes of models of a theory T. If K is a finite field, then Th K has just 1 isomorphism class. If K is an infinite field, then Th K has an isomorphism class of cardinality k, for any infinite cardinality k. Hence we look at the isomorphism classes with the smallest infinite cardinality, the countable cardinality w. We try to find the number of such classes of a theory T. Let n(T) be the number of isomorphism classes of models of cardinality w, of a theory T. We show that if K is a finite field, then n(Th K) = 0. We show that if K is a infinite field, then n(Th K) is not one of 0, 1, 2. We show that if K is an infinite field which is algebraically closed, then n(Th K) = w. We show that if K is an infinite field of characteristic 0 with finite algebraic degree over its prime subfield, then n(Th K) > w. From observations we make, we consider the following possible pattern: if K is a finite field then n(Th K) = 0, if K is an infinite field which is algebraically closed then n(Th K) = w, if K is an infinite field which is not algebraically closed then n(Th K) > w. I do not have a proof of this pattern. We consider the (+,.,0,1,<=)-structures called ordered fields, and show that the order relation (<=) of the rational numbers QQ can be defined by a (+,.,0,1)-formula. At the end we show that QQ and QQ extended with a transcendental element c are not elementary equivalent (+,.,0,1)-structures

Isomorphism theorem for BSS recursively enumerable sets over real closed fields

AbstractThe main result of this paper lies in the framework of BSS computability: it shows roughly that any recursively enumerable set S in RN, N⩽∞, where R is a real closed field, is isomorphic to RdimS by a bijection ϕ which is decidable over S. Moreover the map S↦ϕ is computable. Some related matters are also considered like characterization of the real closed fields with a r.e. set of infinitesimals, and the dimension of r.e. sets