Kurt Gödel facts for kids
Quick facts for kids
Kurt Gödel
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Gödel c. 1926
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| Born |
Kurt Friedrich Gödel
April 28, 1906 |
| Died | January 14, 1978 (aged 71) Princeton, New Jersey, U.S.
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| Citizenship |
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| Alma mater | University of Vienna (PhD, 1930) |
| Known for |
Gödel's incompleteness theorems
Gödel's completeness theorem Gödel's constructible universe Gödel metric (closed timelike curve) Gödel logic Gödel–Dummett logic Gödel's β function Gödel numbering Gödel operation Gödel's speed-up theorem Gödel's ontological proof Gödel–Gentzen translation Gödel–McKinsey–Tarski translation Von Neumann–Bernays–Gödel set theory ω-consistent theory The consistency of the continuum hypothesis with ZFC Axiom of constructibility Compactness theorem Condensation lemma Diagonal lemma Dialectica interpretation Ordinal definable set Slingshot argument |
| Spouse(s) |
Adele Nimbursky
(m. 1938) |
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| Scientific career | |
| Fields | Mathematics, mathematical logic, analytic philosophy, physics |
| Institutions | Institute for Advanced Study |
| Thesis | Über die Vollständigkeit des Logikkalküls (1929) |
| Doctoral advisor | Hans Hahn |
| Influences | |
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Kurt Gödel (born April 28, 1906 – died January 14, 1978) was a very important logician, mathematician, and philosopher. He was born in Austria. Many people consider him one of the most important logicians in history, alongside Aristotle. His ideas greatly changed how people thought about science and philosophy in the 20th century.
Gödel made huge discoveries about the basic rules of mathematics. In 1929, he proved his completeness theorem for his PhD. Two years later, in 1931, he published his famous Gödel's incompleteness theorems. His first incompleteness theorem showed that in any math system strong enough to describe basic numbers, there will always be true statements that cannot be proven or disproven within that system. To do this, he created a method called Gödel numbering. This method turns math statements into numbers. His second theorem, which came from the first, said that a system cannot prove that it is consistent (free from contradictions) using its own rules.
Gödel also showed that two important ideas in math, the axiom of choice and the continuum hypothesis, cannot be proven wrong using the standard rules of Zermelo–Fraenkel set theory. This was a big deal for mathematicians. He also helped us understand how different types of logic are connected.
Contents
Early Life and Learning
Childhood and Family
Kurt Gödel was born on April 28, 1906, in Brünn, which is now Brno in the Czech Republic. His family spoke German. His father, Rudolf Gödel, managed a large textile company. His mother was Marianne Gödel. Kurt's family was active in the culture of Brünn. For example, his grandfather, Joseph Gödel, was a well-known singer.
When he was 12, Gödel became a citizen of Czechoslovakia. This happened after Austria-Hungary broke apart after World War I. Later, he became an Austrian citizen. When Nazi Germany took over Austria in 1938, he became a German citizen. After World War II, in 1948, he became an American citizen.
As a child, Gödel was very curious. His family even nicknamed him "Mr. Why." When he was young, he had rheumatic fever. He got better, but he always worried about his heart. He often had health problems throughout his life.
School Days
Gödel went to a Lutheran school from 1912 to 1916. Then he attended a high school from 1916 to 1924. He was excellent in all his classes, especially math, languages, and religion. At first, he was best at languages, but then he became more interested in history and math. His interest in math grew when his older brother, Rudolf, went to study medicine in Vienna. As a teenager, Gödel studied different subjects, including the writings of Immanuel Kant.
Studying in Vienna
At 18, Gödel joined his brother at the University of Vienna. He already knew a lot of advanced math. He planned to study theoretical physics, but he also took classes in math and philosophy. During this time, he started to believe in mathematical realism, which means he thought math concepts were real. He also joined the Vienna Circle, a group of thinkers.
Gödel became very interested in mathematical logic after attending a seminar about Bertrand Russell's book Introduction to Mathematical Philosophy. He believed mathematical logic was the most important science.
A lecture by David Hilbert about completeness in math systems helped shape Gödel's future work. In 1928, Hilbert and Wilhelm Ackermann asked a big question: "Are the rules of a formal system enough to prove every true statement in that system?"
This question became the focus of Gödel's PhD work. In 1929, at age 23, he finished his dissertation. In it, he proved his completeness theorem for first-order logic. He received his doctorate in 1930.
Career and Discoveries
The Incompleteness Theorems
In 1930, Gödel shared his incompleteness theorems at a conference. He published them in a paper called "[[On Formally Undecidable Propositions of Principia Mathematica and Related Systems|On Formally Undecidable Propositions of Principia Mathematica and Related Systems]]."
In this paper, he showed that for any math system strong enough to describe natural numbers (like Peano axioms):
- If the system is consistent (meaning it doesn't have contradictions), it cannot be complete (meaning it can't prove every true statement).
- The system cannot prove its own consistency.
These theorems were a big shock to mathematicians. They showed that there are limits to what math systems can prove.
The main idea behind the incompleteness theorem is quite simple. Gödel created a math statement that basically says, "This statement cannot be proven in this system." If it could be proven, it would be false, which creates a problem. So, there will always be at least one true statement that cannot be proven within the system. To make this work, Gödel invented Gödel numbering. This method turns statements and proofs into numbers.
Work in the 1930s
In 1932, Gödel became an unpaid lecturer at the University of Vienna. In 1933, Adolf Hitler came to power in Germany. The Nazis' influence grew in Austria. In 1936, Moritz Schlick, who had inspired Gödel's interest in logic, was killed. This event made Gödel very unwell for a time. He needed to take a break to recover.
In 1933, Gödel visited the U.S. for the first time. There, he met Albert Einstein, who became a close friend. Gödel also gave talks about computable functions and the idea of truth in math.
In 1934, Gödel gave lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey. These lectures were later published. He returned to the IAS in 1935. After another period of health issues, he started teaching again in 1937. He worked on proving that the axiom of choice and the continuum hypothesis are consistent with set theory. This means they don't cause contradictions.
On September 20, 1938, he married Adele Nimbursky. His parents had not approved because she was a divorced dancer and six years older than him.
He visited the U.S. again in 1938 and published an important paper. In this paper, he introduced the constructible universe. This is a model of set theory where sets are built from simpler ones. Gödel showed that the axiom of choice and the generalized continuum hypothesis are true in this universe. This meant they were consistent with the main rules of set theory. This discovery was very important for mathematicians.
Life in Princeton
Moving to the U.S.
After Austria became part of Nazi Germany in 1938, Gödel faced problems because of his past connections with Jewish thinkers. The University of Vienna turned down his job application. Also, the German army said he was fit for military service.
To avoid the war, Gödel and his wife left Vienna in 1939. They took the Trans-Siberian railway across Asia, sailed from Japan to San Francisco, and then traveled by train to Princeton. There, Gödel started working at the Institute for Advanced Study (IAS).
Friendship with Einstein
Albert Einstein also lived in Princeton at this time. Gödel and Einstein became very good friends. They often walked together to and from the IAS. No one knew what they talked about during their walks. Einstein once said that his own work didn't mean as much to him anymore, and he came to the Institute just to walk home with Gödel.
In 1947, Einstein and another friend, Oskar Morgenstern, went with Gödel to his U.S. citizenship exam. Gödel had told them he found a flaw in the U.S. Constitution that could lead to a dictatorship. His friends worried he might say something unusual. The judge, Phillip Forman, knew Einstein. When the judge asked Gödel if a dictatorship could happen in the U.S., Gödel started to explain his discovery. The judge quickly moved on, and Gödel became a U.S. citizen.
Gödel became a permanent member of the IAS in 1946. He stopped publishing new papers but continued his work. He became a full professor in 1953 and retired in 1976.
Later Interests
Later in his career, Gödel became interested in philosophy and physics. In 1949, he found solutions to Einstein's field equations that involved closed timelike curves. These solutions suggested that time travel to the past might be possible. This made Einstein question his own theory. These solutions are now called the Gödel metric.
He also studied the works of Gottfried Leibniz. In the 1970s, Gödel shared his own version of Anselm of Canterbury's ontological argument for God's existence. This is known as Gödel's ontological proof.
Awards and Recognition
Gödel received the first Albert Einstein Award in 1951. He also won the National Medal of Science in 1974. He became a member of the American Philosophical Society in 1961 and a Foreign Member of the Royal Society in 1968. The Gödel Prize, an award for important papers in theoretical computer science, is named after him.
Later Life and Death
Later in his life, Gödel experienced periods of poor health. He developed an intense fear of being poisoned. Because of this, he would only eat food prepared by his wife, Adele. When Adele was in the hospital in late 1977, Gödel refused to eat. He became very ill and passed away on January 14, 1978, in Princeton. He was buried in Princeton Cemetery. Adele died in 1981.
Beliefs
Gödel believed in a personal God. He called his philosophy "rationalistic, idealistic, optimistic, and theological." He also believed in an afterlife. He said that if the world is logical and has meaning, then an afterlife must exist.
In a note he never sent, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza." He thought that most religions were "bad," but that the idea of religion itself was not. He also said, "I like Islam: it is a consistent idea of religion and open-minded."
Legacy
Douglas Hofstadter wrote the 1979 book Gödel, Escher, Bach to celebrate the ideas of Gödel, M. C. Escher, and Johann Sebastian Bach. The book explores how Gödel's incompleteness theorem can apply to any computer system, including the human brain.
The Kurt Gödel Society, started in 1987, is named after him. It is an international group that promotes research in logic, philosophy, and the history of mathematics. The University of Vienna has the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has an annual Kurt Gödel lecturer.
Five books of Gödel's collected works have been published. These include his papers, unpublished writings, and letters.
Several books have been written about Gödel's life, including Logical Dilemmas: The Life and Work of Kurt Gödel by John Dawson (2005) and Journey to the Edge of Reason: The Life of Kurt Gödel by Stephen Budiansky (2021). Gödel was also featured in the 2008 BBC documentary Dangerous Knowledge.
See also
In Spanish: Kurt Gödel para niños
- Gödel machine
- Gödel fuzzy logic
- Gödel–Löb logic
- Gödel Prize
- Gödel's ontological proof
- Infinite-valued logic
- List of Austrian scientists
- List of pioneers in computer science
- Mathematical Platonism
- Original proof of Gödel's completeness theorem
- Primitive recursive functional
- Strange loop
- Tarski's undefinability theorem
- World Logic Day
