''Paradoxes of the Infinite'' (German title: ''Paradoxien des Unendlichen'') is a mathematical work by

Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his lib ...

on the theory of sets. It was published by a friend and student,

František Přihonský František () is a masculine given name of Czech origin. It is a cognate of Francis, Francisco, François, and Franz. People with the name include: * Frank Daniel (František Daniel) (1926–1996), Czech film director, producer, and screenwriter ...

, in 1851, three years after Bolzano's death. The work contained many interesting results in

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

. Bolzano expanded on the theme of

Galileo's paradox Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, ''Two New Sciences'', Galileo Galilei made apparently contradictory statements about the positive integers. First, some numbers ...

, giving more examples of correspondences between the elements of an infinite set and

proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...

s of infinite sets. In the work he also explained the term ''Menge'', rendered in English as "set", which he had coined and used in several works since the 1830s.

References

* ''Paradoxes of the Infinite''; trans. by D.A.Steele; London: Routledge, 1950 * (German original) * History of mathematics Infinity Logic literature Set theory 1851 essays Works published posthumously {{mathlogic-stub