从方法论警告到框架性原则:论"有限不可推无限"的逻辑地位 / From Methodological Warning to Framework Principle: On the Logical Status of the Principle of Non-Extrapolation from the Finite to the Infinite

1925年,希尔伯特在《论无限》中指出,“许多对有限成立的定理不能直接和不加限制地推广到无限”。这一论断在现代数学中身份暧昧——它既非由形式系统推导出的定理,亦非系统内部的一阶公理,却在事实上充当了判定无穷问题真伪的元逻辑准则。本文提出,这一困境的根源在于数学界长期将这条论断误读为一个关于“事实”的描述性陈述,而它的实质是一条关于“举证责任”的元方法论原则。 本文的论证分为三步:首先,展示将这一论断解读为一阶逻辑命题——无论是存在性解读还是全称性解读——均无法同时满足逻辑自洽性与实践指导性的要求。其次,论证该论断的真正性质是:“有限域中成立这一事实,本身不构成无限域中成立的充分理由;主张外推的一方负有举证责任”,这是一条约束公理系统建构过程的框架性原则,而非系统内部的一阶公理。最后,本文区分“推演语境”与“辩护语境”,指出 ZFC 在推演层面虽然是严格的一阶演绎系统,但其若干全域公理把在有限集合中具有直觉基础的原则设定为适用于整个集合论宇宙;问题不在于这种设定在系统内部无效,而在于这些全域化选择的元层级辩护并未被充分显式化。 进而,本文以休谟原则与欧几里得原则在无限域中的不相容为例,论证 ZFC 的公理选择包含隐式的理论选择,而非逻辑必然。将这条原则正式确立为一条框架性原则,意味着把 ZFC 建构所依赖的隐式前提从“自然延伸”的叙事中重新识别出来,从而为多元无穷理论体系的公理选择提供一个元方法论框架。 In 1925, Hilbert observed in "On the Infinite" that "many theorems valid for the finite cannot be directly and unrestrictedly extended to the infinite." This statement occupies an ambiguous status in modern mathematics — it is neither a theorem derived within any formal system nor a first-order axiom internal to one, yet it functions de facto as a meta-logical criterion for adjudicating claims about the infinite. This paper argues that the root of this predicament lies in the mathematical community's long-standing misreading of the statement as a descriptive assertion about "facts," whereas its true nature is that of a meta-methodological principle concerning the *burden of proof*. The argument proceeds in three steps. First, we show that interpreting the statement as a first-order logical proposition — whether existentially ("there exists at least one property that cannot be extrapolated," true but too weak) or universally ("no property may be extrapolated by default," logically false) — fails to satisfy simultaneously the requirements of logical coherence and practical guidance. Second, we demonstrate that the genuine character of the statement is: "The mere fact that a proposition holds in the finite domain does not constitute sufficient grounds for its holding in the infinite domain; the party advocating extrapolation bears the burden of proof." This is a *Framework Principle* governing the construction of axiomatic systems, not a first-order axiom internal to one. Third, we distinguish the *context of deduction* from the *context of justification*, showing that while ZFC is a rigorous first-order deductive system at the level of deduction, several of its universal axioms stipulate that principles with intuitive grounding in finite sets apply to the entire set-theoretic universe. The problem is not that such stipulations are invalid within the system, but that the meta-level justification for these universalizations has not been made sufficiently explicit. Taking the incompatibility of Hume's Principle and Euclid's Principle in the infinite domain as a case study, we argue that ZFC's choice of axioms embodies an implicit *theory choice* rather than logical necessity. Explicitly establishing this principle as a Framework Principle means uncovering the implicit premises upon which ZFC's construction depends — extracting them from the narrative of "natural extension" — and thereby providing a meta-methodological framework for the choice of axioms across pluralistic theories of the infinite.