Results 1 
3 of
3
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
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
NorthHolland Publishing Company RECURSMLY TNIVARTANT pRECURSTON THEORY
, 1981
"... 'We int¡oduce recursively invariant precursion theory as a new approach towards recursion theory on an arbitrary limit ordinal É. We follow Friedman and Sacks and call a subset of BBrecursively enumerable if it is.Xrdefinable over Lu. Since FriedmanSacks'notion of a pfinite set is not ..."
Abstract
 Add to MetaCart
'We int¡oduce recursively invariant precursion theory as a new approach towards recursion theory on an arbitrary limit ordinal É. We follow Friedman and Sacks and call a subset of BBrecursively enumerable if it is.Xrdefinable over Lu. Since FriedmanSacks'notion of a pfinite set is not invariant under precursive permutations of p we turn to a different notion. Under all possible invariant generalizations of finite there is a canonical one which we call ifinite. We consider fu¡ther in the inadmissible case those criteria for the adequacy of generalizations of finite which have earlier been developed by Kreisel, Moschovakis and others. We look at infinitary languages over inadmissible sets Lu and the compactness theorem for these languages, the characterization of the basic notions of precursion theory in terms of model theo¡etic invariance, the deflnition of Brecursion theory via an equation calculus and axioms for computation theories. In turns out that in all these approaches the ifinite sets are those subsets of B respectively Lu which behave like finite sets. Invariant Brecursion theory contains classical recursion theory and crecursion theory as special cases. We start in the second half of this paper the systematic development of invariant Brecursion theory for all lfunit ordinals B. We study in particular idegrees, which generalize
@ NorthHolland Publishing Cornpany ON c AND PRECURSMLY ENUMERABLE DEGREES
, 1978
"... Several problems in recursion theory on admissible o¡dinals (arecursion theory) and recursion theory on inadmissible ordinals (Brecursion theory) are studied. Fruitful interactions betweenboth theories are stressed. In the fr¡stpart the admissible collapse is used in order to characterizefor some ..."
Abstract
 Add to MetaCart
(Show Context)
Several problems in recursion theory on admissible o¡dinals (arecursion theory) and recursion theory on inadmissible ordinals (Brecursion theory) are studied. Fruitful interactions betweenboth theories are stressed. In the fr¡stpart the admissible collapse is used in order to characterizefor some inadmissible B the structure of all precursively enumerable degrees asanaccumulationof structures of 2[recursively enumerable degrees for many admissible structures 2[. Thus problems about the Brecursively enumerable degrees can be solved by considering "locally " the analogous problem in an admissible 2 [ (where results of arecursion theory apply). In the second part precursion theory is used as a tool in infinite injurypriority constructions for some particularly interesting c (e.g.,1"). New effects canbeobserved since some structu¡e of the inadmissibleworld above O'isprojected into the arecursivelyenumerable degrees byinvertingthejump. The gained understandingof thejump of arecursively enumerable degrees makes it possible to solve some open problems. A few years ago S.D. Friedman and G.E. Sacks [1] started a new chapter in generalized recursion theory: precursion theory. So far recursion theory was studied only on those initial segments L of the constructible hierarchy where a is admissible. In precursion theory one considers initial segments Lu lor any limit