The Infinity of a theologian and a mathematician. On the history of the controversy of Father Pavel Florensky and Academician Nikolaj N. Luzin

Set theory entered the circle of interests of P.A. Florensky during his years of study at the Mathematical Department of the Physics and Mathematics Faculty of Moscow University. Even then, he became an ardent supporter of the views of George Cantor to infinity. It was Florensky who made the first interpretation of Cantor's ideas in Russian literature [1]. His younger university friend N.N. Luzin became acquainted with set theory somewhat later. Luzin’s ideology was formed under the strong influence of his teachers - Frenchmen E. Borel and A. Lebesgue. While still a student, he became acquainted with the content of the famous dispute about the axiom of arbitrary choice [2, 3] and took the “realist” position towards it. This is evidenced by his correspondence with Florensky [4] (see his letter of 1 (14) May 1906), as well as the text of his famous dissertation, “Integral and trigonometric series” [5]. If for Florensky, who considered infinity from the standpoint of a theologian, no restrictions on the teachings of Cantor are unacceptable (in the dispute about the axiom of Zermelo, the position of the “idealists” is closer to him, but certainly not of the “realists”), then Luzin does not accept such views and seeks to estimate the limits in which the infinite can be used in mathematical reasoning. In his famous “Lectures on analytic sets and their applications” [6], Lusin questioned the right to use even all the irrationality of the Cantor uncountable continuum (the position of a consistent “effectivist”). Father Paul, for whom Cantor's transfinites solve the problem of the "heavenly hierarchy," not only does not seek to avoid contradictions that may arise along this path, but welcomes them as evidence of the "antinomicity of truth." Reference 1.Florensky P.A. On the symbols of infinity (Essay on the ideas of G. Cantor). (In Russian) // Novyj put'. 1904. No. 9. P. 173–235 (reprinted in: Florensky, P.A., Works in 4 volumes. T. 1. Moscow: Mysl'. 1994. P. 79–129). 2. Medvedev F.A. The French school of the theory of functions and sets at the turn of the XIX – XX centuries. (In Russian). Moscow: Nauka. 1976. 3. Medvedev F.A. Early history of the axiom of choice. (In Russian). Moscow: Nauka. 1982. 4. Correspondence N.N. Luzin with P.A. Florensky. Publication and notes by S.S. Demidov, A.N. Parshin, S.M. Polovinkin and P.V. Florensky (In Russian) // Istoriko-matematicheskie issledovaniya. 1989. Vol. 31. Pp. 125–191. 5. Luzin N.N. Integral and trigonometric series (1915) (In Russian) / Collected Works. Vol. 1. Moscow: Publishing House of the USSR Academy of Sciences. 1953. P. 48-212. 6. Luzin N.N. Lectures on analytic sets and their applications (1930) (In Russian) / Collected Works. T. 2. Moscow: Publishing House of the Academy of Sciences of the USSR. 1958. pp. 9–269. See also : Lusin N. Lecons sur les ensembles analytiques et leurs applications.— Paris: Gauthiers-Villars, 1930.— XV + 328 p.