Editing Mathematical logic (section)

=== 19th century ===
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In the middle of the nineteenth century, [[George Boole]] and then [[Augustus De Morgan]] presented systematic mathematical treatments of logic.  Their work, building on work by algebraists such as [[George Peacock (mathematician)|George Peacock]], extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of [[foundations of mathematics]].{{sfnp|Katz|1998|p=686}} In 1847, [[Vatroslav Bertić]] made substantial work on algebraization of logic, independently from Boole.<ref>{{Cite web |title=Bertić, Vatroslav |url=https://www.enciklopedija.hr/clanak/bertic-vatroslav |access-date=2023-05-01 |website=www.enciklopedija.hr}}</ref> [[Charles Sanders Peirce]] later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.

[[Gottlob Frege]] presented an independent development of logic with quantifiers in his ''[[Begriffsschrift]]'', published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until [[Bertrand Russell]] began to promote it near the turn of the century.  The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.

From 1890 to 1905, [[Ernst Schröder (mathematician)|Ernst Schröder]] published ''Vorlesungen über die Algebra der Logik'' in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.

==== Foundational theories ====
Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.

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In logic, the term ''arithmetic'' refers to the theory of the [[natural number]]s. [[Giuseppe Peano]]{{sfnp|Peano|1889}} published a set of axioms for arithmetic that came to bear his name ([[Peano axioms]]), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time [[Richard Dedekind]] showed that the natural numbers are uniquely characterized by their [[mathematical induction|induction]] properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms.{{sfnp|Dedekind|1888}} Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the  recursive definitions of addition and multiplication from the [[successor function]] and mathematical induction.

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In the mid-19th century, flaws in Euclid's axioms for geometry became known.{{sfnp|Katz|1998|p=774}}  In addition to the independence of the [[parallel postulate]], established by [[Nikolai Lobachevsky]] in 1826,{{sfnp|Lobachevsky|1840}} mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert{{sfnp|Hilbert|1899}} developed a complete set of [[Hilbert's axioms|axioms for geometry]], building on [[Pasch's axiom|previous work]] by Pasch.{{sfnp|Pasch|1882}}  The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the [[real line]].  This would prove to be a major area of research in the first half of the 20th century.

The 19th century saw great advances in the theory of [[real analysis]], including theories of convergence of functions and [[Fourier series]]. Mathematicians such as [[Karl Weierstrass]] began to construct functions that stretched intuition, such as [[Continuous, nowhere differentiable function|nowhere-differentiable continuous functions]]. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate.  Weierstrass began to advocate the [[arithmetization of analysis]], which sought to axiomatize analysis using properties of the natural numbers. The modern [[(ε, δ)-definition of limit]] and [[continuous function]]s was already developed by [[Bernard Bolzano|Bolzano]] in 1817,{{sfnp|Felscher|2000}} but remained relatively unknown.
[[Cauchy]] in 1821 defined continuity in terms of [[infinitesimal]]s (see Cours d'Analyse, page 34).  In 1858, Dedekind proposed a definition of the real numbers in terms of [[Dedekind cuts]] of rational numbers, a definition still employed in contemporary texts.{{sfnp|Dedekind|1872}}

[[Georg Cantor]] developed the fundamental concepts of infinite set theory. His early results developed the theory of [[cardinality]] and [[Cantor's first uncountability proof|proved]] that the reals and the natural numbers have different cardinalities.{{sfnp|Cantor|1874}} Over the next twenty years, Cantor developed a theory of [[transfinite number]]s in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the [[Cantor's diagonal argument|diagonal argument]], and used this method to prove [[Cantor's theorem]] that no set can have the same cardinality as its [[powerset]]. Cantor believed that every set could be [[well-ordered]], but was unable to produce a proof for this result, leaving it as an open problem in 1895.{{sfnp|Katz|1998|p=807}}