某些数论的Herbrand相合性
Herbrand Consistency of Some Arithmetical Theories
摘要：
利用对数收缩有界公式的证元的技巧，证明了某些算术理论在有界归纳下的Herbrand相容性的Gödel第二不完全性定理，由于Z. Adamowicz [Herbrand consistency and bounded arithmetic，\textit{Fundamenta Mathematicae} 171（2002）279- 292].在那篇论文中，它表明，一个人不能总是缩小证明的有界公式数学，但在Herbrand的一致性存在，理论${\rm IΔ_0+Ω_m}$与$m\geqslant 2$，任何证明的任何有界公式可以缩短数学。这直接意味着理论$T\supseteq {\rm IΔ_0+Ω_2}$在$T$本身中的Herbrand相容性的不可证明性。本文将上述结果推广到${\rm IΔ_0+Ω_1}$。在裁剪了${\rm IΔ_0}$的Herbrand相容性定义后，我们证明了${\rm IΔ_0}$的相应定理.因此，哥德尔第二不完全性定理的Herbrand版本适用于理论${\rm IΔ_0+Ω_1}$和${\rm IΔ_0}$。
英文摘要：
Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae} 171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ${\rm IΔ_0+Ω_m}$ with $m\geqslant 2$, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory $T\supseteq {\rm IΔ_0+Ω_2}$ in $T$ itself. In this paper, the above results are generalized for ${\rm IΔ_0+Ω_1}$. Also after tailoring the definition of Herbrand consistency for ${\rm IΔ_0}$ we prove the corresponding theorems for ${\rm IΔ_0}$. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories ${\rm IΔ_0+Ω_1}$ and ${\rm IΔ_0}$.
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