THEOREMS OF HYPERARITHMETIC ANALYSIS AND ALMOST THEOREMS OF HYPERARITHMETIC ANALYSIS

James S. Barnes, Jun Le Goh, R. Shore
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Abstract

Abstract Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR $_{0}$ (and so $\Pi _{1}^{1}$ -CA $_{0}$ or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( $\Pi _{1}^{1}$ , r- $\Pi _{1}^{1}$ , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.
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超算术分析的定理和几乎定理
在逆向数学和递归理论的复杂性领域中,超算术分析定理占据了一个不寻常的领域。它们位于图灵跳跃的所有固定(递归)迭代之上,但低于ATR $_{0}$(因此$\Pi _{1}^{1}$ -CA $_{0}$或超跳跃)。证明理论的原理有很长的历史。在这篇论文发表之前,只有一个数学例子。Barnes、Goh和Shore[1]分析了一系列图论中的泛在性定理,这些定理源自Halin[9]关于图中的射线的研究。它们似乎是ACA $_{0}$的典型应用,但实际上是tha。这些结果回答了Montalbán逆向数学开放问题[19]中的问题30,并提供了其他几种不同和不寻常的复杂程度的自然原理。这项工作在[25]中引出了反向数学动物园的一个新领域:超算术分析几乎定理(ATHAs)。当与ACA $_{0}$结合时,它们是tha,但单独使用时非常弱。提供了数学和逻辑的居民。定义了几个保守性类($\Pi _{1}^{1}$, r- $\Pi _{1} $和Tanaka)的推广,并且证明了这些ATHAs以及许多其他原理在RCA $_{0}$上在所有这些意义上是保守的,并且在其他递归理论方式上也是弱的。这些结果回答了Hirschfeldt提出的问题,并在[19]中报告,提供了一长串原则对,其中一个在RCA $_{0}$上非常弱,但在ACA $_{0}$上等效于另一个可能是强的(THA)或非常强的,在一个标准层次结构中,最终比完整的二阶算法更强。
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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