算术理论的图灵-泰勒展开式
Turing-Taylor expansions for arithmetic theories
摘要：
图灵级数经常被用来衡量数学理论的证明理论强度。基于$n$可证明性的图灵级数产生了一个$Π_{n+1}$证明论序数。因此，对于每个理论$U$，我们可以指定相应的$Π_{n+1}$序数序列$\langle |U|_n\rangle_{n>0}$。我们称这个序列为理论的\emph{Turing-Taylor expansion}。在本文中，我们将皮亚诺算术的子理论的图灵-泰勒展开与Ignatiev的多模态可证明性逻辑${\mathbf{GLP}}_ω$的封闭片段的万有模型联系起来。特别是，在第一稿中，我们观察到伊格纳季耶夫模型中的每一点都可以看作是形式数学理论的图灵-泰勒展开。此外，允许图灵-泰勒表达式的皮亚诺算术的每个子理论将定义伊格纳季耶夫模型中的唯一点。
英文摘要：
Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories. Turing progressions based on $n$-provability give rise to a $Π_{n+1}$ proof-theoretic ordinal. As such, to each theory $U$ we can assign the sequence of corresponding $Π_{n+1}$ ordinals $\langle |U|_n\rangle_{n>0}$. We call this sequence a \emph{Turing-Taylor expansion} of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev's universal model for the closed fragment of the polymodal provability logic ${\mathbf{GLP}}_ω$. In particular, in this first draft we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expression will define a unique point in Ignatiev's model.
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