In this paper we analyze excerpts of "Paradoxes of the Infinite", the posthumous work of Bernard Bolzano (1781-1848), in order to show that Georg Cantor's (1845-1918) approach to the problem of defining actual mathematical infinity is not the most natural. In fact, Bolzano's approach to the paradoxes of infinity is more intuitive, while remaining internally coherent. Bolzano's approach, however, had limitations. We discuss implications for teaching, which include a better understanding of the responses of students to situations involving actual mathematical infinity, for it is possible to draw a kind of parallel between these responses and Bolzano's reasoning.