Matthias Schr\"oder 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 Schr\"oder'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.
翻译:Matthias Schr\\"Oder"问了一个问题:在连续版本的Weishrauch Lattice中,是否有最弱的不连续性问题。这样一个问题可以被视为最弱的无法解决的问题。我们引入了不连续性问题,我们证明它完全可以被复制到有效的不连续性问题,以适当的方式定义。然而,从这个意义上讲,Schr\"Otticker的疑问敏感地取决于所选择的逻辑框架,如果我们在Zermelo-Fraenkel 中以依赖性选择和确定性AD的xixix来设置理论,这是一个积极的答案。在Sermelo-Fraenkel 中, 这样一个问题可以被视为最弱的不可解决的问题。 另一方面,我们用完全的绝对性原则来构建不连续的不连续性问题。 我们的确定性概念和不透明性规则的每个问题都是我们所选择的。 我们用瓦德游戏的变的数学问题来证明我们的结果。 存在一个接近的玩家II的策略是这个问题的延续性,但是它也是这个问题的延续性(我们已经通过不确定性概念的简单的不透明性,我们理解了一种不透明的不透明性,我们所理解的不透明性的一个不透明性的问题。