不连续性问题
The Discontinuity Problem
AI总结：
本文研究 Weihrauch 格中是否存在最弱不连续问题，引入不连续性问题并证明其等价于有效不连续问题，结论依赖于公理框架，并利用 Wadge 博弈变体证明。
摘要：
Matthias Schröder 提出了一个问题：在 Weihrauch 格的连续版本中，是否存在最弱的不连续问题。这样的问题可以被视为最弱的不可解问题。我们引入了不连续性问题，并证明了它恰好可归约到以适当方式定义的有效不连续问题。然而，这个答案在何种意义上回答了 Schröder 的问题，敏感地依赖于所选择的公理框架；如果我们在带有依赖选择公理和决定性公理 AD 的 Zermelo-Fraenkel 集合论中工作，则答案是肯定的。另一方面，使用完全选择公理，人们可以构造出不连续但并非有效不连续的问题。因此，Weihrauch 格底部的确切情况敏感地依赖于我们所选择的公理设置。我们使用数学问题的 Wadge 博弈的一个变体来证明我们的结果。虽然玩家 II 存在必胜策略刻画了问题的连续性（正如 Nobrega 和 Pauly 已经表明的），但玩家 I 存在必胜策略则刻画了问题的有效不连续性。我们所说的 Weihrauch 决定性，是指每个问题要么是连续的，要么是有效不连续的这一条件。这种决定性概念是一个相当强的概念，因为它不仅由决定性公理 AD 蕴含，而且还蕴含 Wadge 决定性。我们以对广义生产力概念的简要讨论作为结束。
英文摘要：
Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the continuous version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder's question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work in Zermelo-Fraenkel set theory with dependent choice and the axiom of determinacy AD. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy AD, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity.
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