potential in the mathematical field. (Waldegg 2005, p. 574).

Prior to Bolzano, there were several attempts to find alternative answers, by Grosseteste, Maignan, Rodrigo da Arriaga, and others, cf. the paper *Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Set Theory Inevitable?* (Mancosu 2009). Bolzano, however, went the farthest of all. The main part of Bolzano’s theory of infinity is contained in his last treatise, the *Paradoxes of the Infinite*, which he wrote in 1848, the year of his death. It was published only posthumously in 1851 (Bolzano 1851), (Russ 2004). We proceed primarily from this work, though we take Bolzano’s other books into consideration as well. The paragraph numbers given below refer to the paragraphs in the *Paradoxes of the Infinite*.

Opinions on Bolzano’s theory of infinity
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Bolzano’s theory is usually perceived from the point of view of Cantor’s set theory. The latter is viewed as the only correct theory. Bolzano’s conception thus appears to be either incorrect, or an imperfect predecessor of Cantor’s.

Georg Cantor himself expressed this view, having read Bolzano’s *Paradoxes of the Infinite*. He mentioned them in his 1883 work *Foundations of a General Theory of Manifolds*. While praising Bolzano on 