Gedel o Kantorovom problemu kontinuuma I: Gedelova filozofija matematike
Gödel on Cantor’s Continuum Problem I: Gödel’s Philosophy of Mathematics

Summary/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 first infinite 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 efforts 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 first 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 first 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 finally the epistemological aspect of Gödel's Platonism.

• Details
• Contents