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Real analysis

Cauchy (1789–1857) started the project of giving rigorous bases to infinitesimal calculus. In particular, he rejected the heuristic principle that he called the generality of algebra, which consisted to apply properties of algebraic operations to infinite sequences without proper proofs. In his Cours d'Analyse (1821), he considers very small quantities, which could presently be called "sufficiently small quantities"; that is, a sentence such that "if x is very small then..." must be understood as "there is a (sufficiently large) natural number n such that | x | < 1/ n ". In the proofs he use this in a way that predates the modern (ε, δ)-definition of limit.

The modern (ε, δ)-definition of limits and continuous functions was first developed by Bolzano in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work.

Karl Weierstrass (1815–1897) formalized and popularized the (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as continuous, nowhere-differentiable functions. Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph.

At this point, the program of arithmetization of analysis (reduction of mathematical analysis to arithmetic and algebraic operations) advocated by Weierstrass was essentially completed, except for two points.

Firstly, a formal definition of real numbers were still lacking. Indeed, beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, including Hermann Hankel, Charles Méray, and Eduard Heine, but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means of Dedekind cuts; 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.

The third problem is more subtle: and is related to the foundations of logic: classical logic is a first order logic; that is, quantifiers apply to variables representing individual elements, not to variables representing (infinite) sets of elements. The basic property of the completeness of the real numbers that is required for defining and using real numbers involves a quantification on infinite sets. Indeed, this property may be expressed either as for every infinite sequence of real numbers, if it is a Cauchy sequence, it has a limit that is a real number, or as every subset of the real numbers that is bounded has a least upper bound that is a real number. This need of quantification over infinite sets is one of the motivation of the development of higher-order logics during the first half of the 20th century.