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The embracing of actual infinity in mathematics leads naturally to thequestion of comparing the sizes of infinite collections. The basic dilemma isthat the Cantor Principle (CP), according to which two sets have the same sizeif there is a one-to-one correspondence between their elements, and thePart-Whole Principle (PW), according to which the whole is greater than itspart, are inconsistent for infinite collections. Contemporary axiomaticset-theoretic systems, for instance, ZFC, are based on CP. PW is not valid forinfinite sets. In the last two decades, the topic of sizes of infinite sets hasresurfaced again in a number of papers. A question of whether it is possible tocompare the sizes to comply with PW has been risen and researched. BernardBolzano in his 1848 Paradoxes of the Infinite dealt with principles ofintroducing infinity into mathematics. He created a theory of infinitequantities that respects PW and which is based on sums of infinite series. Weextend Bolzano's theory and create a constructive method for determining theset size of countable sets so that the cardinality of finite sets is preservedand PW is valid. Our concept is close to the numerosity theory from thebeginning of this century but it is simpler and more intuitive. In the results,we partly agree with the theory of numerosities. The sizes of countable setsare uniquely determined, but they are not linearly ordered.

Bolzano and the Part-Whole Principle