Kurt Gödel

=== Studies in Vienna ===
[[File:GoedelKurt.jpg|thumb|upright=1.4|Plaque to Gödel at 43-45 {{ill|Josefstädter Straße|de}}, [[Vienna]], where he discovered his incompleteness theorems]]
At the age of 18, Gödel joined his brother at the [[University of Vienna]]. He had already mastered university-level mathematics.<ref>Dawson 1997, p.&nbsp;24.</ref> Although initially intending to study [[theoretical physics]], he also attended courses on mathematics and philosophy.<ref>At the University of Vienna, Gödel attended mathematics and philosophy courses side by side with [[Hermann Broch]], who was in his early forties. See: {{cite book|url=https://books.google.com/books?id=BFgpBAAAQBAJ&pg=PA27|title=Kurt Kurt Gödel: Das Album |author=Sigmund, Karl|author-link=Karl Sigmund|author2=Dawson Jr., John W.|author-link2=John W. Dawson Jr.|author3=Mühlberger, Kurt|page=27|publisher=Springer-Verlag|year=2007|isbn=978-3-8348-0173-9}}</ref> During this time, he adopted ideas of [[mathematical realism]]. He read [[Immanuel Kant|Kant]]'s {{lang|de|[[Metaphysical Foundations of Natural Science|Metaphysische Anfangsgründe der Naturwissenschaft]]|italic=yes}}, and participated in the [[Vienna Circle]] with [[Moritz Schlick]], [[Hans Hahn (mathematician)|Hans Hahn]], and [[Rudolf Carnap]]. Gödel then studied [[number theory]], but when he took part in a seminar run by [[Moritz Schlick]] which studied [[Bertrand Russell]]'s book ''Introduction to Mathematical Philosophy'', he became interested in [[mathematical logic]]. According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."<ref>Gleick, J. (2011) ''[[The Information: A History, a Theory, a Flood]],'' London, Fourth Estate, p. 181.</ref>

Attending a lecture by [[David Hilbert]] in [[Bologna]] on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and [[Wilhelm Ackermann]] published {{lang|de|Grundzüge der theoretischen Logik|italic=yes}} (''[[Principles of Mathematical Logic]]''), an introduction to [[first-order logic]] in which the problem of completeness was posed: "Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?"<ref name="auto">{{Cite conference
 | title = In the Scope of Logic, Methodology and Philosophy of Science
 | volume = 1|conference=11th International Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999
 | year = 2002|page=291}}
</ref>

This problem became the topic that Gödel chose for his doctoral work.<ref name="auto"/> In 1929, aged 23, he completed his doctoral [[dissertation]] under Hans Hahn's supervision. In it, he established his eponymous [[Gödel's completeness theorem|completeness theorem]] regarding [[first-order logic]].<ref name="auto"/> He was awarded his doctorate in 1930,<ref name="auto"/> and his thesis (accompanied by additional work) was published by the [[Vienna Academy of Science]].