Bernard Bolzano (1781-1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part-whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano's mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano's infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano's views on the mathematical infinite are, after all, consistent.
Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the modeltheoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders's proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind's Habilitationsrede, to which Manders's account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements.
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