Cut elimination for impredicative infinitary systems. Part II ordinal analysis for iterated inductive definitions

In this paper a converse of Barwise's completeness theorem is proved by cut-elimination considerations applied to inductive definitions. We show that among the transitive sets T satisfying some weak closure conditions (closure under primitive-recursive set-functions is more than enough), only the unions of admissible sets satisfy Barwise's completeness theorem in the form stating that if φ ∊ T is a sentence which has a derivation (in the universe) then φ has a derivation in T. See §1 for the origin of the problem in Barwise's paper [Ba].Stated quite briefly the proof is as follows (a step-by-step account including relevant definitions is given in the body of the paper):Let T be a transitive prim.-rec. closed set, and let is nonempty, transitive and closed under pairs}. For each let κ(A) be the supremum of closure ordinals of first-order positive operators on subsets of A (first-order with respect to By Theorem 1 of [BGM], it is enough to prove that rank(T) in order to obtain that T is a union of admissible sets. (The rank of a set is defined by rank(x) = supy ∊ x (rank(y) + 1); since T is prim.-rec. closed, rank(T) = smallest ordinal not in T.)Let We show how to find in T (in fact, in Lω(A)) a derivable sentence τ that has no derivation D such that rank(D) ≤ α. Thus, if τ is to have a derivation in T, rank(T) > α. α is arbitrary (< κ(A)), so rank(T) ≥ κ(A). Q.E.D.

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Cut-elimination for impredicative infinitary systems part I. Ordinal-analysis for ID1

Archive for Mathematical Logic ◽

10.1007/bf02011638 ◽

1981 ◽

Vol 21 (1) ◽

pp. 113-129 ◽

Cited By ~ 10

Author(s):

W. Pohlers

Keyword(s):

Cut Elimination ◽

Ordinal Analysis

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Ordinal analysis of the formal theory for noniterated inductive definitions

Proof Theory - Lecture Notes in Mathematics ◽

10.1007/bfb0083415 ◽

2006 ◽

pp. 109-186

Keyword(s):

Formal Theory ◽

Ordinal Analysis ◽

Inductive Definitions

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Ordinal analysis of non-monotone Π10-definable inductive definitions

Annals of Pure and Applied Logic ◽

10.1016/j.apal.2008.06.014 ◽

2008 ◽

Vol 156 (1) ◽

pp. 160-169

Author(s):

Wolfram Pohlers

Keyword(s):

Ordinal Analysis ◽

Inductive Definitions

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A Model-Theoretic Approach to Ordinal Analysis

Bulletin of Symbolic Logic ◽

10.2307/421195 ◽

1997 ◽

Vol 3 (1) ◽

pp. 17-52 ◽

Cited By ~ 12

Author(s):

Jeremy Avigad ◽

Richard Sommer

Keyword(s):

Second Order ◽

Cut Elimination ◽

Large Set ◽

Theoretic Approach ◽

Natural Numbers ◽

Finite Sets ◽

Ordinal Analysis ◽

Second Order Arithmetic ◽

Combinatorial Properties ◽

Order Arithmetic

AbstractWe describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

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