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in unformalized natural language). See also Cohen 2008. The system of KripkePlatek set theory is closely related to generalized recursion theory. First-order Model Theory by Wilfrid Hodges. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. First-order logic edit Main article: First-order logic First-order logic is a particular formal system of logic. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics ( Ferreirs 2001,. . There are many known examples of undecidable problems from ordinary mathematics. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities (Cantor 1874). 20th century edit In the early decades of the 20th century, the main areas of study were set theory and formal logic. The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution.

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This paper led to the general acceptance of the axiom of choice in the mathematics community. Morley's categoricity theorem, proved by Michael. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be completed. Thus, for example, it is possible to say that an object is a continental drift theory essay whole number using a formula of L1,displaystyle L_omega _1,omega such as (x0 x1 x2).displaystyle (x0)lor (x1)lor (x2)lor cdots. Fevrier to biology (. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien." Translation: "Carrying out this plan by Hilbert for an exposition on proof theory for mathematical logic has experienced an essential delay because, at the stage at which the exposition. Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages.

The CurryHoward isomorphism between proofs and programs relates to proof theory, especially intuitionistic logic. Leopold Kronecker famously stated "God made the integers; all else is the work of man endorsing a return to the study of finite, concrete objects in mathematics. Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming. Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization.