Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s “Completeness Paper” (1930)

EasyChair Preprint 9551

Abstract

The previous Part I of the paper (https://doi.org/10.33774/coe-2022-wlr02) discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit.

Keyphrases: Aristotle, Boolean algebra, Gödel, Husserl, Logicism, Ontology, Pythagoreanism, Russell, bit of information, epsilon calculus, first-order logic, propositional logic, quantum logic, set theory

@booklet{EasyChair:9551,
  author    = {Vasil Penchev},
  title     = {Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödel’s “Completeness Paper” (1930)},
  howpublished = {EasyChair Preprint 9551},
  year      = {EasyChair, 2023}}