There is a well-known gap between metamathematical theorems and their philosophical interpretations. Take Tarski’s Theorem. According to its prevalent interpretation, the collection of all arithmetical truths is not arithmetically definable. However, the underlying metamathematical theorem merely establishes the arithmetical undefinability of a set of specific Gödel codes of certain artefactual entities, such as infix strings, which are true in the standard model. That is, as opposed to its philosophical reading, the metamathematical theorem is formulated (and proved) relative to a specific choice of the Gödel numbering and the notation system. A similar observation applies to Gödel’s and Church’s theorems, which are commonly taken to impose severe limitations on what can be proved and computed using the resources of certain formalisms. The philosophical force of these limitative results heavily relies on the belief that these theorems do not depend on contingencies regarding the underlying formalisation choices. The main aim of this paper is to provide metamathematical facts which support this belief. While employing a fixed notation system, I showed in previous work (Review of Symbolic Logic, 2021, 14(1):51–84) how to abstract away from the choice of the Gödel numbering. In the present paper, I extend this work by establishing versions of Tarski’s, Gödel’s and Church’s theorems which are invariant regarding both the notation system and the numbering. This paper thus provides a further step towards absolute versions of metamathematical results which do not rely on contingent formalisation choices.

元数学定理与其哲学解释之间存在着众所周知的差距。以塔斯基定理为例。根据其普遍的解释，所有算术真理的集合在算术上是不可定义的。然而，底层的元数学定理仅仅建立了某些人工实体（例如中缀字符串）的一组特定哥德尔码的算术不可定义性，这在标准模型中是正确的。也就是说，与其哲学解读相反，元数学定理是相对于哥德尔编号和符号系统的特定选择来制定（和证明）的。类似的观察也适用于哥德尔定理和丘奇定理，这些定理通常被认为对使用某些形式主义的资源可以证明和计算的内容施加了严格的限制。这些限制性结果的哲学力量在很大程度上依赖于这样的信念：这些定理不依赖于潜在的形式化选择的偶然性。本文的主要目的是提供支持这一信念的元数学事实。在使用固定符号系统时，我在之前的工作（符号逻辑评论，2021，14（1）：51-84）中展示了如何从哥德尔编号的选择中抽象出来。在本文中，我通过建立塔斯基、哥德尔和丘奇定理的版本来扩展这项工作，这些定理在符号系统和编号方面都是不变的。因此，本文向不依赖于偶然形式化选择的元数学结果的绝对版本又迈出了一步。