In his table of judgements, Kant added infinity as a third quality. An infinite judgement ‘All S are non-P’ is said to differ from the affirmative ‘All S are P’ because it ascribes a negative predicate; and it differs from the negative ‘No S is P’ because it has a richer content. The present paper puts three interpretations of this surplus content to six tests. Among other things, it is examined whether these interpretations marry up with Kant’s solution to the (...) first antinomy, his conception of analyticity and the principle of complete determination. The unpleasant conclusion is that none of the interpretations can make its mark as the most adequate one. It thus remains unclear what Kant had in mind when positioning infinite judgements besides affirmative and negative ones. (shrink)

In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in order to clarify Ernst Zermelo’s reading of Kant’s antinomy, according to which the latter is deeply rooted in the (...) tendency of the mind of producing “creative progress” and “inclusive closure”. The aim of this paper is to explain in which sense and why Kant’s treatment of the antinomies attracts the attention of Zermelo in the early 1900s and which aspects of his second axiomatic system have been inspired by Kant’s philosophy. Thus, by reading Kant’s antinomy ‘through Zermelo’s eyes’—with emphasis on the concept of regressive series in indefinitum and on that of regressive series ad infinitum – this paper identifies the echoes of Kant’s work in the making of the ZFC set theory. (shrink)