Ranked structures and arithmetic transfinite recursion
نویسندگان
چکیده
منابع مشابه
Ranked Structures and Arithmetic Transfinite Recursion
ATR0 is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the “well-founded part” of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian p-groups, and countabl...
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The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than ~0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic techniques. A variety of additional results tha...
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We consider in this paper a special class of predicates. Every class predicate, say p, has one argument denoted Y of type list. All the other arguments are integers and form a vector denoted X. The predicates are primitive recursively deened over the structure of Y, and the auxiliary predicates are arithmetic relations over the integer arguments. When two atoms of this class, say p 1 (Y, X 1) a...
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Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles ...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2008
ISSN: 0002-9947
DOI: 10.1090/s0002-9947-07-04285-7