Theory Ordinals Can Replace ZFC in Computer Science

Abstract

The theory Ordinals can serve as a replacement for the theory ZFC because:
• Ordinals are a very well understood mathematical structure.
• There is only one model of Ordinals up to a unique isomorphism, which decides every proposition of the theory Ordinals in the model.
• The theory Ordinals is much more powerful than ZFC. Standard mathematics that has been carried out in ZFC can more easily be done in Ordinals. Axioms of ZFC are in effect theorems of Ordinals. The type Ordinal is larger than any set in ZFC because no set can be placed in one-to-one correspondence with instances of the type Ordinal.
• The theory Ordinals is algorithmically inexhaustible, i.e., it is impossible to computationally enumerate theorems of the theory thereby reinforcing the intuition behind [Franzén, 2004]. Contrary to [Church 1934], the conclusion in this article is to abandon the assumption that theorems of a theory must be computationally enumerable while retaining the requirement that proof checking must be computationally decidable.
• There are no “monsters” [Lakatos 1976] in models of Ordinals such as the ones in models of 1st-order ZFC. Consequently unlike ZFC, the theory Ordinals is not subject to cyberattacks using “monsters” in models such as the ones that plague 1st-order ZFC.
• The theory Ordinals formally proves its own consistency contra [Gödel 1931]
• Longstanding issues such as the nonexistence of large cardinals and the nonexistence of nonstandard models [Cohen 1966] are resolved in the theory Ordinals.

The theory Ordinals is based on intensional types as opposed to extensional sets of ZFC. Using intensional types together with strongly-typed ordinal induction is key to proving that there is just one model of the theory Ordinals up to a unique isomorphism.

Keywords: Uniquely Categorical Theories, Strong Types, Scalable Intelligent Systems, Actor Model of Computation, Sergei Artemov, Jeremy Avigad, Steve Awodey, Jon Barwise, Nicolas Bourbaki, Cesare Burali-Forti, John Burgess, Alonzo Church, Paul Cohen, Thierry Coquand, Haskell Curry, Dedekind, Jean-Yves Girard