巨人与虚幻巨人古德斯坦原理
Giant and illusionary giant Goodstein principles
摘要：
我们分析了几个自然的古德斯坦原则，其本身是相对于阿克曼函数和扩展的阿克曼函数定义。这些阿克曼函数是由不超过$\varepsilon_0$的序数标记的良好建立的规范快速增长函数。在所考虑的古德斯坦原理中，巨原理在理论上是强有力的（在阿克曼情形下，在$\mathrm{PA}$中是不可证明的，在扩展的阿克曼情形下，在$\mathrm{ID}_1$中是不可证明的），而其他的巨原理，即虚幻的巨原理，虽然乍看上去很强，但结果会相对弱得多。
英文摘要：
We analyze several natural Goodstein principles which themselves are defined with respect to the Ackermann function and the extended Ackermann function. These Ackermann functions are well established canonical fast growing functions labeled by ordinals not exceeding $\varepsilon_0$. Among the Goodsteinprinciples under consideration, the giant ones, will be proof-theoretically strong (being unprovable in $\mathrm{PA}$ in the Ackermannian case and being unprovable in $\mathrm{ID}_1$ in the extended Ackermannian case) whereas others, the illusionary giant ones, will turn out to be comparatively much much weaker although they look strong at first sight.
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