Bernard Bolzano 1781 1848 Studied philosophy mathematics and

Bernard Bolzano (1781 -1848) • Studied philosophy, mathematics and physics in Prague starting 1796, doctorate 1804. • Decided to become a priest was ordained in 1804 and took up a chair in the philosophy of religion. • His sermons were against government policy and he was dismissed. He then turned back to mathematics. He worked on Cauchy sequences - as they are called today. • Worked on Set-Theory: Paradoxien des Unendlichen 1851 • Most famous theorem: The theorem of Bolzano. Weierstrass: every bounded sequence of real numbers contains a convergent subsequence. The intermediate value theorem is related to this theorem which he also proved. • His works were however not widely known.

Bernard Bolzano (1781 -1848) A First Analysis of the Infinite • A first treaty on the infinite which does not discard it as an undesirable source for paradox. • Exploited the tool of 1 -1 correspondence. • For infinite sets the idea of 1 -1 correspondence is not well behaved with respect to inclusion. • There are 1 -1 correspondences between sets and proper subsets of them.

Bolzano’s Paradoxes of the infinite • § 19. Not all infinite sets are equal with respect to their multiplicity – One could say that all infinite sets are infinite and thus one cannot compare them, but most people will agree that an interval in the real line is certainly a part and thus agree to a comparison of infinite sets. • § 20 There are distinct infinite sets between which there is 1 -1 correspondence. It is possible to have a 1 -1 correspondence between an infinite set and a proper subset of it. – y=12/5 x and y=5/12 x gives a 1 -1 correspondence between [0, 5] and [0, 12]. • § 21 If two sets A and B are infinite, one can not conclude anything about the equality of the sets even if there is a 1 -1 correspondence. – If A and B are finite and A is a subset of B such that there is a 11 correspondence, then indeed A=B – The above property is thus characteristic of infinite sets.