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Post’s Problem for Ordinal Register Machines
"... Abstract. We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinitetime Turing machines, and also provides ..."
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Abstract. We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinitetime Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.
Atomic models higher up
, 2008
"... There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ωCK 1 and where the LωCK 1,ωtheory of M is not ωcategorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ωtheory. Most wellknown models with unboundedtype; such a atoms below ω CK 1 al ..."
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There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ωCK 1 and where the LωCK 1,ωtheory of M is not ωcategorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ωtheory. Most wellknown models with unboundedtype; such a atoms below ω CK 1 also realize a nonprincipal L ω CK 1,ω will have Scott rank ωCK 1 + 1. Makkai ([4]) produces a hyperarithmetical model of Scott rank ωCK 1 whose LωCK 1,ωtheory is ωcategorical. A computable variant of Makkai’s example is produced in [2]. model that preserves the Σ1admissibility of ω CK 1
The theory of the metarecursively enumerable degrees
"... Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equival ..."
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Abstract. Sacks [Sa1966a] asks if the metarecursivley enumerable degrees are elementarily equivalent to the r.e. degrees. In unpublished work, Slaman and Shore proved that they are not. This paper provides a simpler proof of that result and characterizes the degree of the theory as O (ω) or, equivalently, that of the truth set of L ω CK
The Role of True Finiteness in the Admissible Recursively Enumerable Degrees
 Memoirs of the American Mathematical Society
"... Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In ..."
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Abstract. When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of αfiniteness. As examples we discuss both codings of models of arithmetic into the recursively enumerable degrees, and nondistributive lattice embeddings into these degrees. We show that if an admissible ordinal α is effectively close to ω (where this closeness can be measured by size or by cofinality) then such constructions may be performed in the αr.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the firstorder language of partially ordered sets, and so these results also show that there are natural elementary differences between the structures of αr.e. degrees for various classes of admissible ordinals α. Together with coding work which shows that for some α, the theory of the αr.e. degrees is complicated, we get that for every admissible ordinal
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
I „ DEFINABLE SETS WITHOUT Z „ INDUCTION
"... Abstract. We prove that the FriedbergMuchnik Theorem holds in all models of Zj collection under the base theory P ~ + TLo. Generalizations to higher dimensional analogs are discussed. We also study the splitting of r.e. sets in these weak models of arithmetic. The study of recursively enumerable (r ..."
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Abstract. We prove that the FriedbergMuchnik Theorem holds in all models of Zj collection under the base theory P ~ + TLo. Generalizations to higher dimensional analogs are discussed. We also study the splitting of r.e. sets in these weak models of arithmetic. The study of recursively enumerable (r.e.) sets is one of the focal points of modern recursion theory, with a history going back to the investigations initiated by Post. Post's problem asks whether there exists a nonrecursive incomplete r.e. set over co. This problem is answered affirmatively by the FriedbergMuchnik Theorem which asserts the existence of two r.e. sets having incomparable Turing degrees. The method of priority argument was introduced for the solution, and since then the technique has been developed and refined to a level of extraordinary sophistication and complexity. The FriedbergMuchnik Theorem has been generalized in many directions. Sacks and Simpson [ 14] proved this for all admissible ordinals, and Friedman [7] obtained it for many inadmissibe ordinals. Also, in [3] we show that a related and natural version of Post's problem has a positive solution for all limit ordinals and that the same holds when relativized to 0 ' for all admissible ordinals. In another direction, Simpson (unpublished) has shown that the priority argument used for the solution holds in all models of fragments of Peano arithmetic satisfying Ei induction, and Slaman and Woodin [15] have shown that in all models of I.x collection there is a positive solution to the original Post problem. The more difficult question of whether the FriedbergMuchnik Theorem holds in these models was left open. In this paper we study some basic problems of r.e. sets in the setting of firstorder arithmetic. We work in the usual firstorder language of arithmetic with an additional function symbol for exponentiation. Let P ~ + 7Zn denote the set of Peano axioms (including the definition of exponentiation), yet with the full induction scheme replaced by induction restricted to Xn formulas. Let BZ„ and H,n denote respectively the E „ collection and the £ „ induction scheme. Paris and Kirby [13] proved that over the base theory P ~ + I2Z0, 7X „ implies Received by the editors September 6, 1990.
THE THEORY OF THE α DEGREES IS UNDECIDABLE
"... Abstract. Let α be an arbitrary Σ1admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are αdegrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of αdegrees is undecidabl ..."
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Abstract. Let α be an arbitrary Σ1admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are αdegrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of αdegrees is undecidable. 1.
Ordinal Machines and Admissible Recursion Theory
"... We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theor ..."
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We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: αcomputability theory. We compare the new theory to αrecursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of αcomputable and αrecursive as well as αcomputably enumerable and αrecursively enumerable completely agree. Moreover there is an isomorphism of parts of the degree structure induced by αcomputability and of a degree structure in αrecursion theory, which allows us to transfer, e.g., the SacksSimpson theorem or Shore’s density theorem to αcomputability theory. We emphasize the algorithmic approach by giving a proof of the SacksSimpson theorem, which is solely based on αmachines and does not rely on constructibility theory.