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**11 - 17**of**17**### Recursive In A Generic Real

- J. Symbolic Logic
, 2000

"... . There is a comeager set C contained in the set of 1-generic reals and a rst order structure M such that for any real number X, there is an element of C which is recursive in X if and only if there is a presentation of M which is recursive in X. 1. Introduction The theory of a generic real is the ..."

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. There is a comeager set C contained in the set of 1-generic reals and a rst order structure M such that for any real number X, there is an element of C which is recursive in X if and only if there is a presentation of M which is recursive in X. 1. Introduction The theory of a generic real is the theory of the almost everywhere behavior of all of the reals, and as such it can be well approximated. Consequently, constructions which can be implemented relative to any generic real can usually be simulated by approximation. For example, if a set of natural numbers X is recursive in every element of a co-meager set, then X is recursive. Similar statements for arithmetic in or constructible from are equally valid. Counter to these observations, Slaman [1] produced a rst order structure M such that for all reals X, X is not recursive if and only if there is a presentation of M which is recursive in X. X's computing a presentation of M gives an existential criterion for determining whet...

### THE SLAMAN-WEHNER THEOREM IN HIGHER RECURSION THEORY

"... A central concern of computable model theory is the restriction that algebraic structure imposes on the information content of an object of study. One asks about a countable object, what information is coded intrinsically into this object, which cannot be avoided by passing to an isomorphic copy of ..."

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A central concern of computable model theory is the restriction that algebraic structure imposes on the information content of an object of study. One asks about a countable object, what information is coded intrinsically into this object, which cannot be avoided by passing to an isomorphic copy of the object? Given a

### COMPUTABLY ENUMERABLE PARTIAL ORDERS

"... Abstract. We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly s ..."

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Abstract. We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly stronger than the latter. We then show that every ∅ ′-computable structure (or even just of c.e. degree) has the same degree spectrum as some computably enumerable (co-c.e.) partial order, and hence that there is a c.e. (co-c.e.) partial order with spectrum equal to the set of nonzero degrees. 1.

### unknown title

"... 3.1 Turing degrees of isomorphism types of structures................... 21 ..."

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3.1 Turing degrees of isomorphism types of structures................... 21

### TURING DEGREES OF NONABELIAN GROUPS

"... Abstract. For a countable structure A, the (Turing) degree spectrum of A is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of A has the least degree d, thenwesay that d is the (Turing) degree of the isomorphism type of A. Sofar, degrees of the isomorphism types have b ..."

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Abstract. For a countable structure A, the (Turing) degree spectrum of A is the set of all Turing degrees of its isomorphic copies. If the degree spectrum of A has the least degree d, thenwesay that d is the (Turing) degree of the isomorphism type of A. Sofar, degrees of the isomorphism types have been studied for abelian and metabelian groups. Here, we focus on highly nonabelian groups. We show that there are various centerless groups whose isomorphism types have arbitrary Turing degrees. We also show that there are various centerless groups whose isomorphism types do not have Turing degrees.

### Russian Academy of Sciences, Siberian Branch

, 2010

"... We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the ..."

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We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order Q. New results include realizations of the set of nonlown Turing degrees as the spectrum of a relation on Q for all n ≥ 1, and a realization of the set of all nonlown Turing degrees as the spectrum of a linear order whenever n ≥ 2. The state of current knowledge is summarized in a table in the concluding section.