We present an abstract development of Gödel's incompleteness theorems, performed with the help of the Isabelle/HOL proof assistant. We analyze sufficient conditions for the theorems' applicability to a partially specified logic. In addition to the usual benefits of generality, our abstract perspective enables a comparison between alternative approaches from the literature. These include Rosser's variation of the first theorem, Jeroslow's variation of the second theorem, and the Świerczkowski–Paulson semantics-based approach. As part of our framework's validation, we upgrade Paulson's Isabelle proof to produce a mechanization of the second theorem that does not assume soundness in the standard model, and in fact does not rely on any notion of model or semantic interpretation.
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