Mathematical Sciences: Proof-Theoretical Investigations of Theories

Rathjen打算致力于以下四个问题及相关问题。(1)对PI-1-2理解进行了序贯分析。(2)在递归大序数的基础上发展强序数表示系统，以代替它们的大基数类似物。(3)确定弱集合理论中限制基的证明论强度。(4)关于算术语句，Feferman理论T-0和单调算子的最小不动点公理比T-0保守吗？证明论涉及公理系统和其中的证明方法，特别是不同系统的相对强度。如果在B中有证明的每个命题在A中也有证明，则称系统A适当强于系统B，但反之亦然。调查人员正在沿着这些思路探索一些问题，技术难度和专家们感兴趣的程度各不相同，但没有一个问题真的可以用外行的话进行简短的讨论。

Rathjen intends to work on the following four problems and related issues. (1) Give an ordinal analysis of Pi-1-2 comprehension. (2) Develop strong ordinal representation systems on the basis of recursively large ordinals in lieu of their large cardinal analogues. (3) Determine the proof-theoretic strength of restricted foundations in weak set theories. (4) Is Feferman's theory T-0 together with least fixed point axioms for monotone operators conservative over T-0 with respect to arithmetic sentences? Proof theory concerns itself with axiomatic systems and methods of proof within them, particularly with the relative strength of different systems. The system A is said to be properly stronger than the system B if every proposition that has a proof in B also has a proof in A but the converse is not so. The investigator is pursuing a number of questions along these lines, varying in technical difficulty and degree of interest to the experts, but none is really susceptible to a brief discussion in laymen's terms.