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This paper examines a proof of incompleteness by Stanisław Świerczkowski in a paper entitled “Finite sets and Gödel’s incompleteness theorems”. Świerczkowski’s proof is different to most other proofs of incompleteness because the formal system that is used for the proof is a system of hereditarily finite sets. Świerczkowski claims that this makes his proof simple and elegant, and that it has enabled his proof to be complete and without any gaps or omissions, nor relying on references to other publications. Świerczkowski claims that for this reason his proof his superior to most other proofs of incompleteness. However, the author makes an elementary error in his proof that renders the proof invalid; his proof relies on a confusion regarding the domain of the variables involved. Świerczkowski bases his proof on an assumption that a certain formula has a certain domain, whereas a logical analysis demonstrates that this is not the case.
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