基于参数化ackermann-pÉter函数的Goodstein序列

T. Arai, S. Wainer, A. Weiermann
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引用次数: 1

摘要

继我们的[6]之后,虽然使用了一些不同的方法,但我们用参数化ackermann - psamter函数引入了Goodstein序列的进一步变体。每个序列都被证明是终止的,并且这些事实的证明理论强度通过序数赋值来校准,从而产生一系列理论的独立性结果:PRA, PA, $\Sigma ^1_1$ -DC $_0$, ATR $_0$,直到ID $_1$。关键是所谓的证明论边界函数的“Hardy层次”,它提供了将goodstein型序列与正整数的参数化范式表示相关联的统一方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
GOODSTEIN SEQUENCES BASED ON A PARAMETRIZED ACKERMANN–PÉTER FUNCTION
Abstract Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA, $\Sigma ^1_1$ -DC $_0$ , ATR $_0$ , up to ID $_1$ . The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal form representations of positive integers.
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