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Biographical Notes: Kurt Gödel
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[5. Biographical Notes.]


Kurt Gödel.

The logician, mathematician and philosopher Kurt Gödel (Brno 1906 - Princeton, NJ, 1978) studied mathematics and physics at Vienna University, and received his doctorate under Hans Hahn with a dissertation ("Die Vollständigkeit der Axiome des logischen Funktionenkalküls" in Monatshefte für Mathematik und Physik, 37, 1930) in which he proved the completeness of first order logic. In Künigsberg congress (1930) Gödel announced that he has proved the incompleteness of formal arithmetic; the proof was published the following year in the article "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme" in Monatshefte für Mathematik und Physik, 38, 1931.
Gödel worked also on set theory and non-classical logics, such as intuitionistic and modal logic. He proved that the continuum hypothesis is consistent with the axioms of classical set theory.
He was interested in the mathematical aspects of the theory of relativity, and proved the existence of solutions of Einstein's relativistic equations in which time travels in the past are possible. These solutions describes so-called "rotating universes" with cosmological constant different from zero. "By making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space" ("A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy" in Albert Einstein. Philosopher-Scientist, edit by P. A. Schilpp, "The Library of Living Philosopher", Evanston, Ill., 1949; quoted from Gödel, Kurt, Collected Works, vol.2, Oxford: Oxford University Press, 2001).
Gödel philosophic attitudes seems in contrast with those usually attributed to the other logical positivists. Of particular interest is his position on the problem of the foundations of mathematics. Gödel asserted that mathematics is not reducible to formal logic; on the contrary, the proper method of mathematics is "the intuitive grasping of ever newer axioms that are logically independent from the earlier ones" (Gödel, Kurt, "The modern development of the foundations of mathematics in the light of philosophy" in Collected Works, vol. 3, Oxford: Oxford University Press, 2001). The mathematician attempts to solve every clearly posed mathematical questions through the conscious extension of the axioms of mathematics; thus new axioms, which are not logically derivable from the already established axioms, become evident. This method, based on intuition and not on formal logic, "agrees in principles with the Kantian conception of mathematics" (ibid.).
Gödel defended an idealistic philosophy of time in his contributions to Albert Einstein. Philosopher-Scientist. In "rotating universes" there is the possibility of time travels; therefore there is no possibility to define in this universe an objective time. Thus, according to Gödel, we have to recognized that time is not an objective entity; on the contrary, we have to adopt Kantian analysis of time, and consider "change as an illusion or an appearance due to our special mode of perception" ("A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy" in Collected Works, vol.2, Oxford: Oxford University Press, 2001).

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