- Kateřina Trlifajová
title: 'Bolzano’s Infinite Quantities'
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Introduction
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Existence of an actual infinity
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There are two major questions that anyone dealing with actual infinity has to address:

1.  Is there any actual infinity?

2.  If so, are there multiple infinities? How can we compare them?

An answer to the first question has been sought by many eminent thinkers, philosophers and mathematicians, starting from Aristotle, who formulated the question, to Cantor (Šebestík 1992). Aristotle rejected actual infinity, and traditional scholasticism responded similarly in the famous thesis of *Infinitum actu non datur.* It was not until the end of the 19th century that Georg Cantor introduced actual infinity in mathematics in the form of infinite sets. He gave a theological justification for their existence, placing them in the mind of God (Dauben 1979), (Halett 1986). A characteristic excerpt from Cantor’s letter to Jeiler from 1895, quoted in (Halett 1986, p. 21), says “in particular, there are transfinite cardinal numbers and transfinite ordinal types which, just as much as the finite numbers and forms, possess a definite mathematical uniformity, discoverable by men. All these particular modes of the transfinite have existed from eternity as ideas in the Divine intellect.” Once 