Kurt Gödel

Kurt Friedrich Gödel (, ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher.

Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an effect upon scientific and philosophical thinking in the 20th century, a time when 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 Richard Dedekind, Georg Cantor and Gottlob Frege.

Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic), into the German-speaking family of Rudolf Gödel (1874–1929), the managing director and part owner of a major textile firm, and Marianne Gödel (née Handschuh, 1879–1966).

At the time of his birth the city had a German-speaking majority which included his parents.

His father was Catholic and his mother was Protestant and the children were raised as Protestants.

The ancestors of Kurt Gödel were often active in Brünn's cultural life.

For example, his grandfather Joseph Gödel was a famous singer in his time and for some years a member of the Brünner Männergesangverein (Men's Choral Union of Brünn).

Gödel automatically became a citizen of Czechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the First World War.

According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".

Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion.

Although Gödel had first excelled in languages, he later became more interested in history and mathematics.

His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna, where he attended medical school at the University of Vienna.

During his teens, Gödel studied Gabelsberger shorthand, and criticisms of Isaac Newton, and the writings of Immanuel Kant.

At the age of 18, Gödel joined his brother at the University of Vienna.

He had already mastered university-level mathematics.

Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy.

During this time, he adopted ideas of mathematical realism.

He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, 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."

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 Grundzüge der theoretischen Logik (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?"

This problem became the topic that Gödel chose for his doctoral work.

Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931.

The first incompleteness theorem states that for any ω-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.

To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.

Gödel 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 former result opened the door for mathematicians to assume the axiom of choice in their proofs.

He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

In February 1929, he was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship.

In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision.

In it, he established his eponymous completeness theorem regarding first-order logic.

He was awarded his doctorate in 1930, and his thesis (accompanied by additional work) was published by the Vienna Academy of Science.

"Kurt Gödel's achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement."

When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32.

In 1948, after World War II, at the age of 42, he became an American citizen.

In his family, the young Gödel was nicknamed Herr Warum ("Mr. Why") because of his insatiable curiosity.

According to his brother Rudolf, at the age of six or seven, Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire life.