论文标题
与古德斯坦一起散步
A walk with Goodstein
论文作者
论文摘要
古德斯坦的原则可以说是第一个纯粹的数字理论陈述，已知独立于Peano算术。它涉及自然数序列，最初似乎会很快增长，但最终减少到零。这些序列是根据基于自然数的指示系统定义的。在本文中，我们探讨了对此类符号系统的最优性概念，并将其应用于经典的Goodstein过程，基于乘法而不是指示的较弱的变体，以及基于Ackermann函数的更强变体。特别是，我们介绍了基础变化最大的概念，并展示了它如何导致古德斯坦结果的深远扩展。
Goodstein's principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to grow very quickly, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we explore notions of optimality for such notation systems and apply them to the classical Goodstein process, to a weaker variant based on multiplication rather than exponentiation, and to a stronger variant based on the Ackermann function. In particular, we introduce the notion of base-change maximality, and show how it leads to far-reaching extensions of Goodstein's result.
