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Israel Journal of Mathematics

, Volume 29, Issue 2–3, pp 221–238 | Cite as

Inductive definitions, models of comprehension and invariant definability

  • K. R. Apt
Article
  • 33 Downloads

Abstract

The connections between inductive definability and models of comprehension are studied. Let Open image in new window = 〈A, R l, ...,R n 〉 be an infinite structure and letI φ be a set inductively defined by a formulaφ of the second order language Open image in new window . We prove that if Open image in new window is a model of Δ 1 1 -Comprehension relativized toφ, andφ is Open image in new window -absolute, then for everyη smaller than the height of Open image in new window (h( Open image in new window )),I φ is in Open image in new window . If Open image in new window is aβ-structure which satisfies Σ 1 1 -Comprehension relativized toφ and WF(X), and φ is Open image in new window -absolute, thenI φ is in Open image in new window and ‖φ| <h ( Open image in new window ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models1 of Δ 1 1 -Comprehension.

Keywords

Free Variable Order Structure Inductive Relation Order Language Inductive Definition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    K. R. Apt,ω-models in analytical hierarchy, Bull. Acad. Polon. Sci.20 (1972), 901–904.MATHMathSciNetGoogle Scholar
  2. 2.
    Y. N. Moschovakis,Elementary Induction on Abstract Structures, North-Holland, Amsterdam, 1974.MATHGoogle Scholar
  3. 3.
    Y. N. Moschovakis,On monotone inductive definability, Fund. Math.82 (1974), 39–83.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1978

Authors and Affiliations

  • K. R. Apt
    • 1
  1. 1.Mathematical CentreAmsterdamThe Netherlands

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