He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. GГ¶del also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo-Fraenkel set theory, assuming that its axioms are consistent. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency. To prove this, GГ¶del developed a technique now known as GГ¶del numbering, which codes formal expressions as natural numbers. The first incompleteness theorem states that for any П‰-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms. GГ¶del published his first incompleteness theorem in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, GГ¶del had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege. Kurt Friedrich GГ¶del ( / Л€ ЙЎ ЙњЛђr d Й™l/ GUR-dЙ™l, German: ( listen) April 28, 1906В вЂ“ January 14, 1978) was a logician, mathematician, and philosopher. Гњber die VollstГ¤ndigkeit des LogikkalkГјls (On the Completeness of the Calculus of Logic) (1929) Mathematics, mathematical logic, analytic philosophy, physics

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