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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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P.R.-regulated systems of notation and the subrecursive hierarchy equivalence property
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by Fred Zemke PDF
Trans. Amer. Math. Soc. 234 (1977), 89-118 Request permission

Abstract:

We can attempt to extend the Grzegorczyk Hierarchy transfinitely by defining a sequence of functions indexed by the elements of a system of notation $\mathcal {S}$, using either iteration (majorization) or enumeration techniques to define the functions. (The hierarchy is then the sequence of classes of functions elementary in the functions of the sequence of functions.) In this paper we consider two sequences ${\{ {F_s}\} _{s \in \mathcal {S}}}$ and ${\{ {G_s}\} _{s \in \mathcal {S}}}$ defined by iteration and a sequence ${\{ {E_s}\} _{s \in \mathcal {S}}}$ defined by enumeration; the corresponding hierarchies are $\{ {\mathcal {F}_s}\} ,\{ {\mathcal {G}_s}\} ,\{ \mathcal {E}{_s}\}$. We say that $\mathcal {S}$ has the subrecursive hierarchy equivalence property if these two conditions hold: (I) ${\mathcal {E}_s} = {\mathcal {F}_s} = {\mathcal {G}_s}$ for all $s \in \mathcal {S}$; (II) ${\mathcal {E}_s} = {\mathcal {E}_t}$ for all $s,t \in \mathcal {S}$ such that $|s| = |t|(|s|$ is the ordinal denoted by s). We show that a certain type of system of notation, called p.r.-regulated, has the subrecursive hierarchy equivalence property. We present a nontrivial example of such a system of notation, based on Schütte’s Klammersymbols. The resulting hierarchy extends those previously in print, which used the so-called standard fundamental sequences for limits $< {\varepsilon _0}$.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 234 (1977), 89-118
  • MSC: Primary 02F20; Secondary 02F35
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0498045-0
  • MathSciNet review: 0498045