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Bernard Bolzano
(1781-1848)
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• 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 BolzanoWeierstrass: 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=12/5x and y=5/12x 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.