On Abyss Wagers, 5 of 5

VI. Epilogue

Gödel’s Second Incompleteness Theorem states that, due to the paradoxes of self-reference, an arithmetical deduction system is consistent, if and only if it cannot prove its consistency.

Gödel’s Second Incompleteness Theorem implies that the validity of arithmetical reasoning – and by extension, all reasoning – cannot be guaranteed within reason itself. Therefore reason must be taken on faith.

This article argues that it is reasonable to do so, by an argument akin to Pascal’s Wager.

Footnotes

*Nathaniel Hellerstein is Adjunct Instructor of Mathematics at City College of San Francisco in San Francisco, California. He is an iconoclastic logician by trade and inclination, and author of books such as “Diamond – A Paradox Logic”, World Scientific Series on Knots and Everything, Volume 23 (2010).

**Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", Monatshefte für Mathematik und Physik, v. 38 n. 1, pp. 173–198.

—, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in Solomon Feferman, ed., 1986. Kurt Gödel Collected works, Vol. I. Oxford University Press, pp. 144–195. ISBN 978-0195147209. The original German with a facing English translation, preceded by an introductory note by Stephen Cole Kleene.

—, 1951, "Some basic theorems on the foundations of mathematics and their implications", in Solomon Feferman, ed., 1995. Kurt Gödel Collected works, Vol. III, Oxford University Press, pp. 304–323. ISBN 978-0195147223.