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<p>In this paper, we prove that for any computable numberings <i>β &lt; α</i> of an arbitrary family of c.e. sets, where α is equivalent to the completion of some noncomplete numbering, there exists a chain of its computable numberings (with respect to the reducibility of numberings) bounded above by the numbering α and having <i>β</i> as its least element whose order type is the first nonconstructive ordinal. We also establish that this statement is true for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sum_{n}^{0}-\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> </mrow> <mn>0</mn> </msubsup> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation> computable numberings (<i>n &gt;</i> 1) <i>β &lt; α</i> provided that <i>α is</i> ∅<sup>(<i>n−</i>1)</sup>-precomplete.</p>

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One Property of Complete Numberings of Computable and Generalized Computable Families

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • M. Kh. Faizrahmanov

摘要

In this paper, we prove that for any computable numberings β < α of an arbitrary family of c.e. sets, where α is equivalent to the completion of some noncomplete numbering, there exists a chain of its computable numberings (with respect to the reducibility of numberings) bounded above by the numbering α and having β as its least element whose order type is the first nonconstructive ordinal. We also establish that this statement is true for any \(\sum_{n}^{0}-\) ∑ n 0 - computable numberings (n > 1) β < α provided that α is ∅(n−1)-precomplete.