 Chronologically, Bernard Bolzano was the first, but he exerted almost no influence. The high quality of his work in logic and the foundations of mathematics is well known. A book entitled Paradoxien des Unendlichen was posthumously published in 1851. Here Bolzano argued in detail that a host of paradoxes surrounding infinity are logically harmless, and mounted a forceful defence of actual infinity. He proposed an interesting argument attempting to prove the existence of infinite sets, which bears comparison with Dedekind’s later argument (1888). Although he employed complicated distinctions of different kinds of sets or classes, Bolzano recognized clearly the possibility of putting two infinite sets in one-to-one correspondence, as one can easily do, e.g., with the intervals \([0, 5]\) and \([0, 12]\) by the function \(5y = 12x\). However, Bolzano resisted the conclusion that both sets are “equal with respect to the multiplicity of their parts” [1851, 30–31]. In all likelihood, traditional ideas of measurement were still too powerful in his way of thinking, and thus he missed the discovery of the concept of cardinality (however, one may consider Non-Cantorian ideas, on which see Mancosu 2009). 