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

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

Notes on sum-tests and independence tests

We study statistical sum-tests and independence tests, in particular for computably enumerable semimeasures on a discrete domain. Among other things, we prove that for universal semimeasures every Sigma0/1-sum-test is bounded, but unbounded Pi0/1-sum-tests exist, and we study to what extent the latter can be universal. For universal semimeasures, in the unary case of sum-test we leave open whether universal Pi0/1-sum-tests exist, whereas in the binary case of independence tests we prove that they do not exist