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Gödel Mathematics Versus Hilbert Mathematics. I. the Gödel Incompleteness (1931) Statement: Axiom or Theorem?

EasyChair Preprint no. 9073

56 pagesDate: October 24, 2022

Abstract

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: weather it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic. The main argument consists in the contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate: correspondingly, by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. The axiom of choice transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. The Gödel incompleteness statement relies on the contradiction of the axioma of induction and infinity.

Keyphrases: Boolean algebra, completeness, dual axiomatics, Euclidean and non-Euclidean geometries, Fifth postulate of Euclid, finitism, foundations of mathematics, Gödel, Hilbert arithmetic, Hilbert Program, Husserl, incompleteness, Logicism, Peano arithmetic, Phenomenology, Principia Mathematica, propositional logic, Riemann space curvature, Russell, set theory

BibTeX entry
BibTeX does not have the right entry for preprints. This is a hack for producing the correct reference:
@Booklet{EasyChair:9073,
  author = {Vasil Penchev},
  title = {Gödel Mathematics Versus Hilbert Mathematics. I. the Gödel Incompleteness (1931) Statement: Axiom or Theorem?},
  howpublished = {EasyChair Preprint no. 9073},

  year = {EasyChair, 2022}}
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