不是其任何形式表示的标准解释的递归数论关系的某些结果
Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal representations
摘要：
我们给出了一个精确的定义，一个正式的数学对象作为任何符号的一个单独的常数，谓词字母，或一个函数字母，可以通过定义引入到一个正式的数学语言，而不会引起不一致。然后，我们显示：有一个初等的、递归的数论关系，它不是一个形式的数学对象，因为它不是哥德尔形式系统P中任何表示的标准解释;递归数论函数的值域并不总是定义一个（递归可枚举集）形式数学对象在任何对P建模的公理化集合论中是一致的;不存在P-公式Con（P），其标准解释明确地等价于Goedel的数论定义“P是相容的”;每个递归数论函数在P中不是强可表示的;塔斯基对“可满足性”和“真”的定义可以是建设性的，直观上是无可非议的，通过建设性地重新表述Church的论文;图灵机的经典定义可以扩展到包括自终止、收敛和振荡例程;一个建设性的丘奇命题意味着，首先，每一个部分递归数论函数可以建设性地扩展为一个唯一的总函数，其次，我们可以有效地定义非经典图灵可计算的可计算函数;图灵和康托的对角线方法不一定定义柯西序列。
英文摘要：
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting inconsistency. We then show: there is an elementary, recursive, number-theoretic relation that is not a formal mathematical object, since it is not the standard interpretation of any of its representations in Goedel's formal system P; the range of a recursive number-theoretic function does not always define a (recursively enumerable set) formal mathematical object consistently in any Axiomatic Set Theory that models P; there is no P-formula, Con(P), whose standard interpretation is unambiguously equivalent to Goedel's number-theoretic definition of "P is consistent"; every recursive number-theoretic function is not strongly representable in P; Tarski's definitions of "satisfiability" and "truth" can be made constructive, and intuitionistically unobjectionable, by reformulating Church's Thesis constructively; the classical definition of Turing machines can be extended to include self-terminating, converging, and oscillating routines; a constructive Church's Thesis implies, firstly, that every partial recursive number-theoretic function can be constructively extended as a unique total function, and, secondly, that we can define effectively computable functions that are not classically Turing-computable; Turing's and Cantor's diagonal arguments do not necessarily define Cauchy sequences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
