GÖDEL ON CANTOR’S CONTINUUM PROBLEM I: GÖDEL’S PHILOSOPHY OF MATHEMATICS
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Abstract
Cantor’s continuum problem is the question: How many points are there
on a line in Euclidean space? Cantor (Georg Cantor) believed that the number of
points on a line, i.e. the cardinality of the continuum, is the fi rst infi nite cardinal that
comes after the cardinality of the set of natural numbers. This is a rough formulation
of Cantor’s continuum hypothesis. Cantor repeatedly tried to prove the continuum
hypothesis during his career. However, his eff orts did not bear fruit. At the beginning
of the 20th century, set theory was formulated as an axiomatic theory. That axiomatic
theory is called Zermelo-Frenkel set theory with axiom of choice, i.e., ZFC set theory.
Based on the results of Kurt Gödel and Paul Cohen, we know that the ZFC theory
neither proves nor disproves Cantor's hypothesis, leaving Cantor's continuum problem
unsolved. However, according to Gödel, Cantor's continuum problem can be solved
by extending ZFC set theory with new axioms that will give us a more complete
insight into the structure of the universe of sets. This paper is the fi rst in a series of
two papers in which we aim to present some of Gödel’s proposals for an approach
to solving Cantor’s continuum problem, as well as Gödel’s Platonist philosophical
position underlying those proposals. This paper is focused on presenting Gödel's
philosophy of mathematics. To that end, we will fi rst present representative positions
in the philosophy of mathematics and their main representatives. Then we will explain
what Hilbert’s program is and what, according to Gödel, are the consequences of
impossibility of its implementation. Then we deal with the ontological and fi nally the
epistemological aspect of Gödel's Platonism.
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References
Adžić, M., Gedel o aksiomatizaciji teorije skupova, Doktorska disertacija, Beograd
Barker, S., Filozofija matematike, Nolit, Beograd 1973.
Boolos, G., „The Iterative Conception of Set”, The Journal of Philosophy br. 68 (8).
Brouwer, L. E. J. , „Intuitionism and formalism” u Philosophy of mathematics: Selected
readings (priredili P. Benacerraf i H. Putnam.), Cambridge University Press,
Cambridge 1983.
Brouwer, L. E. J., „Consciousness, philosophy, and mathematics” u Philosophy
of mathematics: Selected readings (priredili P. Benacerraf i H. Putnam.),
Cambridge University Press, Cambridge 1983.
Frege, G., „O smislu i nominatumu” u Ogledi o jeziku i značenju (priredili A. Pavković
i Ž. Lazović), Filozofsko društvo Srbije, Beograd 1992.
Frege, G., „Osnove aritmetike: Logičko-matematičko istraživanje pojma broja” u
Osnove aritmetike i drugi spisi (priredili F. Grgić i M. Hudoletnjak Grgić),
Kruzak, Zagreb 1995.
Frege, G., „Funkcija i pojam” u Osnove aritmetike i drugi spisi (priredili F. Grgić i M.
Hudoletnjak Grgić), Kruzak, Zagreb 1995.
Gödel, K., „Russell's mathematical logic” u Kurt Gödel. Collected works. Volume II.
Publications 1938 – 1974. (priredili S. Feferman, J. W. Dawson Jr., S. C.
Kleene, G. H. Moore, R. M. Solovay i J. van Heijenoort), Oxford University
Press, Oxford 1990.
Gödel, K., „What is Cantor's continuum problem? 1947” u Kurt Gödel. Collected
works. Volume II. Publications 1938 – 1974. (priredili S. Feferman, J. W.
Dawson Jr., S. C. Kleene, G. H. Moore, R. M. Solovay i J. van Heijenoort),
Oxford University Press, Oxford 1990.
Gödel, K. „What is Cantor's continuum problem? 1964” u Kurt Gödel. Collected
works. Volume II. Publications 1938 – 1974. (priredili S. Feferman, J. W.
Dawson Jr., S. C. Kleene, G. H. Moore, R. M. Solovay i J. van Heijenoort),
Oxford University Press, Oxford 1990.
Gödel, K., „Some basic theorems on the foundations of mathematics and their
implications” u Kurt Gödel. Collected works. Volume III. Unpublished
Essays and Lectures. (priredili S. Feferman, J. W. Dawson Jr., W. Goldfarb,
C. Parsons i R. Solovay), Oxford University Press, Oxford 1995.
Hilbert, D., „On the Infinite” u From Frege to Gödel: A Source Book in Mathematical
Logic, 1879 – 1931 (priredio J. van Heijenoort), Harvard University Press,
Massachusetts 1967.
Linnebo, Ø., Philosophy of Mathematics, Princeton University Press, Princeton 2017.
Maddy, P., „Believing the Axioms. I”, The Journal of Symbolic Logic, br. 53 (2).
Mijajlović, Ž., Marković, Z. i Došen, K., Hilbertovi problemi i logika, Zavod za
udžbenike i nastavna sredstva, Beograd 1986.
Quine, W. V. O., Philosophy of Logic, Harvard University Press, Cambridge 1986.
Russell, B. Principles of Mathematics, Routledge, London 2010.
Wang, H. „The concept of set” u Philosophy of mathematics: Selected readings
(priredili P. Benacerraf i H. Putnam.), Cambridge University Press,
Cambridge 1983.
Horsten, L., „Philosophy of Mathematics”, https://plato.stanford.edu/archives/
win2023/entries/philosophy-mathematics/ 10.10.2024.