论文标题
一致性操作员的必要迭代
Evitable iterates of the consistency operator
论文作者
论文摘要
让我们修复一个合理的子系统$ t算法；为什么$ t $的自然扩展是按一致性强度预先订购的？在以前的工作中，提出了一种方法。这项工作的目的是将相对于$ t $的Lindenabum代数单调的递归功能进行分类。根据一个乐观的猜想，大致上，每个这样的功能都必须等同于一致性运算符的迭代$ \ mathsf {con} _t^α$。 在先前的工作中，作者建立了这种乐观猜想的第一个案例；粗略地，每个递归单调函数要么与限制中的身份算子一样弱，要么与限制中的$ \ mathsf {consf {consf {con} _t $一样强。然而，在本说明中，我们证明了这种乐观的猜想在下一步中已经失败了。递归单调函数既不如限制中的$ \ MATHSF {con} _t $弱，也不如$ \ Mathsf {con} _t _t^2 $在限制中。实际上，对于每$α$，我们都会产生一个偶有功能，其功能与$ \ m mathsf {con}^α_t$相同，但既稳定地却像$ \ mathsf {con} _t $一样弱。
Let's fix a reasonable subsystem $T$ of arithmetic; why are natural extensions of $T$ pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^α$ of the consistency operator in the limit. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $α$, we produce a function that is cofinally as strong as $\mathsf{Con}^α_T$ yet cofinally as weak as $\mathsf{Con}_T$.
