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A RECURSIONTHEORETIC CHARACTERIZATION OF CONSTRUCTIBLE REALS
"... Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in Lr The notion was introduced by Boolos and Putnam [1] who also initiated the study (from the recursiontheoretic viewpoint) of what is known today as " ..."
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Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in Lr The notion was introduced by Boolos and Putnam [1] who also initiated the study (from the recursiontheoretic viewpoint) of what is known today as &quot; the fine structure of L&quot;. In the settheoretic context, Jensen [3] later extended their results to all levels of the constructible universe. Still, as all the constructible reals occur in LWIL (CO1 L is the first constructibly uncountable ordinal), a great amount of insight into their structure can be gained by studying this &quot; small &quot; universe. One may cite as examples the fruitful investigations by various people on the Turing and hyperdegrees of these reals, the basis theorems associated with them, and so on. Yet another approach is to study fragments Ly for y < cot L. The general program is: study the structure of Ly when (a) y is not an index, and (b) y is an index. Leeds and Putnam [4] and also Marek and Srebrny [5] have taken up (a). Among other things, they showed that if y is the limit of indices but is itself not an index, then Ly is a model of ZF minus the power set axiom, and is hereditarily countable. The first result related to (b) was obtained by Boolos and
Sets which do not have subsets of every higher degree
 Journal of Symbolic Logic
, 1978
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The
Models with High Scott Rank
, 2008
"... Scott rank is a measure of modeltheoretic complexity; the Scott rank of a structure A in the language L is the least ordinal β for which A is prime in its Lωβ,ωtheory. By a result of Nadel, the Scott rank of a structure A is at most ωA 1 + 1, where ωA 1 is the least ordinal not recursive in A. We ..."
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Scott rank is a measure of modeltheoretic complexity; the Scott rank of a structure A in the language L is the least ordinal β for which A is prime in its Lωβ,ωtheory. By a result of Nadel, the Scott rank of a structure A is at most ωA 1 + 1, where ωA 1 is the least ordinal not recursive in A. We say that the Scott rank of A is high if it is at least ωA 1. Let α be a Σ1 admissible ordinal. A structure A of high Scott rank (and for which ω A 1 = α) will have Scott rank α + 1 if it realizes a nonprincipal Lα,ωtype, and Scott rank α otherwise. For α = ω CK 1, the least nonrecursive ordinal, several sorts of constructions are known. The Harrison ordering ω CK 1 (1 + η), where η is the ordertype of the rationals, has Scott rank ω CK 1 + 1. Makkai constructs a model with Scott rank ω CK 1 whose L ω CK 1,ωtheory is ℵ0categorical. Millar and Sacks produce a model A with Scott rank ω CK 1 (in which ω A 1 = ω CK 1) but whose L ω CK 1,ωtheory is not