  The embracing of actual infinity in mathematics leads naturally to the
question of comparing the sizes of infinite collections. The basic dilemma is
that the Cantor Principle (CP), according to which two sets have the same size
if there is a one-to-one correspondence between their elements, and the
Part-Whole Principle (PW), according to which the whole is greater than its
part, are inconsistent for infinite collections. Contemporary axiomatic
set-theoretic systems, for instance, ZFC, are based on CP. PW is not valid for
infinite sets. In the last two decades, the topic of sizes of infinite sets has
resurfaced again in a number of papers. A question of whether it is possible to
compare the sizes to comply with PW has been risen and researched. Bernard
Bolzano in his 1848 Paradoxes of the Infinite dealt with principles of
introducing infinity into mathematics. He created a theory of infinite
quantities that respects PW and which is based on sums of infinite series. We
extend Bolzano's theory and create a constructive method for determining the
set size of countable sets so that the cardinality of finite sets is preserved
and PW is valid. Our concept is close to the numerosity theory from the
beginning 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 sets
are uniquely determined, but they are not linearly ordered.
