Maximum schemes in arithmetic. (English) Zbl 0810.03046
The maximum principle for arithmetic says that a definable function with a bounded domain has a bounded range. Formal versions of this principle can be written in many forms, one of which is the collection principle saying that a definable function whose domain is an initial segment with a top has a bounded range, and strong collection principle saying that a partial definable function defined on an initial segment with a top has a bounded range. In the paper these, and other formal versions of the maximum principle, relativized to the levels of the arithmetical hierarchy, are defined, compared, and placed within the well-known diagrams comparing the proof-theoretic strength of the induction and collection schemes. Proofs are elementary.
Reviewer: R.Kossak (New York)
Keywords:
Peano arithmetic; collection scheme; induction scheme; maximum principle; proof-theoretic strengthReferences:
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