section_path: History/Modern analysis
section_name: Modern analysis
paragraph_text: Modern analysis
  
  The developers of calculus used real numbers and limits without defining them rigorously. In his Cours d'Analyse (1821), Cauchy made calculus rigorous, but he used the real numbers without defining them, and assumed without proof that every Cauchy sequence has a limit and that this limit is a real number.
  
  In 1854 Bernhard Riemann highlighted the limitations of calculus in the method of Fourier series, highlighting the need for a rigorous definition of the real numbers.
  
  Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, including Hermann Hankel, Charles Méray, and Eduard Heine, leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes of Cauchy sequences. Several problems were left open by these definitions, which contributed to the foundational crisis of mathematics. Firstly both definitions suppose that rational numbers and thus natural numbers are rigorously defined; this was done a few years later with Peano axioms. Secondly, both definitions involve infinite sets (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's set theory was published several years later. Thirdly, these definitions imply quantification on infinite sets, and this cannot be formalized in the classical logic of first-order predicates. This is one of the reasons for which higher-order logics were developed in the first half of the 20th century.
  
  In 1874, Cantor showed that the set of all real numbers is uncountably infinite, but the set of all algebraic numbers is countably infinite. Cantor's first uncountability proof was different from his famous diagonal argument published in 1891.