David Hilbert argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created." Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics.

Skolem, Thoralf (1920 "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen Videnskapsselskapet Skrifter,.The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the Entscheidungsproblem is algorithmically unsolvable.The 19th century saw great advances in the theory pandora file recovery mac and images of real analysis, including theories of convergence of functions and Fourier series.Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post.Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard ( 1905 ) discovered Richard's paradox.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.The study of computability came to be known as recursion theory, because early formalizations by Gödel and Kleene relied on recursive definitions of functions.Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers.Algebraic logic Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics.

Cambridge, Mass: Harvard University Press, isbn, (pbk.) Hilbert, David (1899 Grundlagen der Geometrie, Leipzig: Teubner, English 1902 edition ( The Foundations of Geometry ) republished 1980, Open Court, Chicago.His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities (Cantor 1874).Contemporary research in set theory includes the study of large cardinals and determinacy.In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset.It does not encompass intuitionistic, modal or fuzzy logic.Ferreirós ( 2001 ) surveys the rise of first-order logic over other formal logics in the early 20th games zuma star wars 2009 gratis century.