按不动点对角化
Diagonalizing by Fixed-Points
摘要：
对角化的一个普遍模式是由N. S. Yanofsky（2003），其中存在一个（对角化和矛盾的）对象意味着存在某个函数的不动点。许多自指悖论和对角证明的定理都可以在这个图式中找到。在这里，我们在对角化的通用模式中拟合更多的定理，例如欧几里得关于素数的无限性的定理和Boolos（1997）关于一个集合与其幂集的非等数量性的Cantor定理的新证明。然后，在线性时态逻辑中，我们证明了在这种逻辑中不存在不动点，其证明类似于Yablo悖论的论证。这样，雅布洛悖论第一次在线性时态逻辑的框架内成为一个真正的数理逻辑定理。证明中再次使用了文中的对角图式，并证明了G。Priest的封闭图式（1997）可以适合我们的普遍对角/定点图式。我们还表明存在的支配（阿克曼样）功能（支配一个给定的可数集的功能-像原始递归）使用的模式。
英文摘要：
A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema. Here, we fit more theorems in the universal schema of diagonalization, such as Euclid's theorem on the infinitude of the primes and new proofs of Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset. Then, in Linear Temporal Logic, we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox. Thus, Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic. Again the diagonal schema of the paper is used in this proof, and also it is shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal/fixed-point schema. We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions---like primitive recursives) using the schema.
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