We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence \(\alpha \) which extends a weak arithmetical theory (which we take to be \({{\,\mathrm{I\Delta _{0}+\exp }\,}}\)) such that for some formula \(\Theta \) and any arithmetical sentence \(\varphi \), \(\Theta (\ulcorner \varphi \urcorner )\equiv \varphi \) is provable in \(\alpha \). We say that a sentence \(\beta \) is definable in a sentence \(\alpha \), if there exists an unrelativized translation from the language of \(\beta \) to the language of \(\alpha \) which is identity on the arithmetical symbols and such that the translation of \(\beta \) is provable in \(\alpha \). Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not \(\Sigma _2\)-definable in the standard model of arithmetic. We conclude by remarking that no \(\Sigma _2\)-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.

我们研究由算术语言的真值定义的可定义性关系导出的偏序结构。形式上，真理的定义是扩展弱算术理论的任何句子\(\alpha \)（我们将其视为\({{\,\mathrm{I\Delta _{0}+\exp }\,} }\) ) 使得对于某些公式\(\Theta \)和任何算术句子\(\varphi \)，\(\Theta (\ulcorner \varphi \urcorner )\equiv \varphi \)可以在\(\ α \）。我们说句子\(\beta \)在句子\(\alpha \)中是可定义的，如果存在从\(\beta \)语言到\(\alpha \)语言的非相对化翻译，其中是算术符号上的恒等式，因此\(\beta \)的翻译可以在\(\alpha \)中证明。我们的主要结果是，由相对于基本算术理论保守的真值定义组成的结构形成了可数的普适分配格。此外，我们概括了 Pakhomov 和 Visser 的结果，表明真理的定义（哥德尔代码）集在算术标准模型中不可定义\(\Sigma _2\) 。我们得出的结论是，满足某些进一步的自然条件的\(\Sigma _2\)句子不能成为算术语言的真理定义。