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<p>The note concerns the comparative study of two approaches to generalized computability over the real numbers: computable analysis and Σ-definability in hereditarily finite su-perstructures. As known, there exist total computable functions that are not Σ-definable and discontinuous Σ-definable functions that are not computable. In this note, we show that even among continuous real functions, there exists a Σ-definable function that is not computable in the sense of computable analysis. The proof uses a diagonalization construction with infinite computable disjunctions of Δ<sub>0</sub>-formulas.</p>

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Continuity in Computable Analysis and Σ-Definability

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • S. A. Aleksandrova,
  • N. A. Bazhenov

摘要

The note concerns the comparative study of two approaches to generalized computability over the real numbers: computable analysis and Σ-definability in hereditarily finite su-perstructures. As known, there exist total computable functions that are not Σ-definable and discontinuous Σ-definable functions that are not computable. In this note, we show that even among continuous real functions, there exists a Σ-definable function that is not computable in the sense of computable analysis. The proof uses a diagonalization construction with infinite computable disjunctions of Δ0-formulas.