Based on this paradox, Galileo claimed that the attributes of “equal”, “greater” and “smaller” could not be applied to infinite, but only to finite quantities. Similarly, Leibniz refused to compare infinite collections in size, although he did not reject actual infinity *per se*.[^3]

Cantor based his set theory on the second principle. The one-to-one correspondence is decisive: from its existence, one can derive the “size” of sets – Cantor calls it the *cardinality.* The part-whole principle then fails dramatically, for all infinite sets.

Bolzano built the foundations of his theory of infinity on the first principle. Here is a pertinent comment by Guillerrmina Waldegg, who investigated the approach of students to infinity and to comparison of infinite sets:

> Bolzano’s criterion, based on the part-whole correspondence is more “intuitive” because it is nearer to concrete (finite) experience (in fact it corresponds to the mainstream of the historical moment, and to common sense in all times). On the other hand, a one-to-one relationship is less visible. Cantor’s solution could only be reached by means of a total detachment from meaning and intuition; his solution is therefore more problematic and difficult to understand, but it was well accepted once it proved its 