Gödel 在 1931 年证明的两条不完备性定理表明：只要一个形式化数学系统足够强，能够表达算术，就不可能既完备又一致地捕捉全部数学真理。Raatikainen 强调，这意味着对正整数 1、2、3…… 的全部真理并不能从任意有限组公理中推出；一些问题在原则上就超出当前方法。Jouko Väänänen 将其概括为一种“Gödel barrier”：表达力越强，证明效力往往越弱。
围绕连续统假设（CH）的争论显示了这一点。CH 断言实数集合是继自然数集合之后的第二小无限集，但它在标准公理下不可判定；加上不同的新公理，CH 既可被证明为真，也可被证明为假。Claus Kiefer 进一步指出，这种不可判定性也牵涉物理学，因为物理定律使用数学语言，连续时空中的无穷多个点会在广义相对论中导致奇点，并在标准模型里产生需借助重整化消除的无穷量。
分歧在于这是否意味着数学或物理的终结。Rebecca Goldstein、Rachael Alvir 和 Juliette Kennedy 认为，Gödel 并未推翻 Hilbert Program 的全部抱负，而只是否定了“有限公理、有限步骤即可穷尽一切真理”的设想；未来也许可通过更强公理、无限公理序列，或全新的逻辑系统来决定更多命题。无论如何，不完备性已成为数学中的基本事实：它限制了机械化形式化，却也显示数学比 Hilbert 的有限主义图景更宽广。
Gödel’s 1931 incompleteness theorems show that any formal mathematical system rich enough to express arithmetic cannot be both complete and consistent. Raatikainen stresses that the full truth about positive integers 1, 2, 3… does not follow from any finite set of axioms, so some questions are, in principle, beyond current methods. Jouko Väänänen frames this as a “Gödel barrier”: the stronger a logic’s expressive power, the weaker its effectiveness in proving statements.
The debate becomes concrete in the continuum hypothesis (CH). CH says the set of real numbers is the second-smallest infinite set after the natural numbers, yet it is undecidable in standard axioms; with different extra axioms, it can be made true or false. Claus Kiefer argues that this matters for physics because laws are written in mathematical language, and a continuum of points in spacetime creates singularities in general relativity and infinities in the Standard Model that require nonintuitive renormalization.
The disagreement is whether this implies a limit on mathematics or merely on one formal framework. Rebecca Goldstein, Rachael Alvir, and Juliette Kennedy argue that Gödel did not destroy the Hilbert Program in every sense; he rejected the dream that a finite list of axioms and finite deduction steps could settle everything. Future progress might come from stronger axioms, an infinite hierarchy of axiomatic systems, or an entirely new logic. In that view, incompleteness is a permanent feature of formalization, but also evidence that mathematics is wider than Hilbert’s finitist ideal.