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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
nonclassical 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 socalled "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.
PhilosopherScientist, 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.
PhilosopherScientist. 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).
