Recrafting Foundations of Computer Science

A Great Power strategic cyberattack can take down a country’s economy and keep it down for an indefinite period using reiterated cyberattacks [Sanger and Mazzetti 2016; Greenberg 2019]. Millions of people could die as a result. An Apollo-scale project is required in this decade to create and deploy technology sufficient to prevent successful strategic cyberattacks. Foundation for needed technology is over-viewed in this article.Intelligent Systems are characterized by integration of empirical information in close human interaction by systems that are educable, adaptive-in-real-time, and self-informative (cf. [Perry 2012]). Intelligent Systems will enable cyberattacks that could cause even more economic, political, and social damage than what has already been inflicted. A foundational cyberattack is an attack over the Internet that attacks the foundations by which a system acquires and uses information.“Theorems are Enumerable” has been a central theorem in foundations of mathematics. Based on theorem enumerablity, Alonzo Church discovered a bug in the then current theory of computation that he didn't find a way to fix [Church 1934]. The bug opens up possible cyberattacks, which can be prevented using a much more powerful theory of computation based on "Actors" as explained in this article. Other foundational cyberattacks are possible against Intelligent Systems using many-core computations of a kind that cannot be performed by a nondeterministic Turing machine. Consequently, such computations are outside the applicability of the Church/Turing theory of computation thereby requiring use of the more powerful theory based on Actors. Technology and practices recommended in this article can prevent many foundational cyberattacks. Development and universal deployment of Intelligent Systems in this decade will pose huge new challenges for cyberdefenseBecause foundational theories need to be both powerful and flexible, they are right at the edge of inconsistency. Consequently, they must be protected using strong technical means. Foundations are difficult because interactions among possible alternatives can be obscure and there is no way to mathematically prove which system to adopt. Contra McQueen, foundational rules are not to be demolished. Instead, they must be carefully recrafted. “Monster” is a term introduced in [Lakatos 1976] for a mathematical construct that introduces inconsistencies and paradoxes. Euclid, Richard Dedekind, Gottlob Frege, Bertrand Russell, Kurt Gödel, Ludwig Wittgenstein, Alonzo Church, Alan Turing, and Stanisław Jaśkowski all had issues with mathematical monsters as discussed in this article. Monsters can lurk long undiscovered. For example, that “theorems are provably computational enumerable” [Euclid approximately 300 BC] is a monster was only discovered after millennia when [Church 1934] used it to identify fundamental inconsistency in the foundations of mathematics that is resolved in this article.This article explains how the ActorsTheory and Ordinals recraft Hilbert's thesis was that every proposition is either provable or disprovable every mathematical proposition is, that is, inferentially decidable [Hilbert 1930]. Others doubted inferential decidability held. It would mean that whether a proposition is a theorem could be algorithmically decided. [Gödel 1931] seemed to have settled the issue of inferential undecidability using the proposition I'mUnprovable. Contra [Gödel 1931], Wittgenstein soon correctly proved that existence of I'mUnprovable leads to inconsistency in foundations. His proof was ignored and subsequently derided by Gödel. The proposition I'mUnprovable is inconsistent with foundational theorems including “a theorem can be used in subsequent proofs”. The method used to construct I'mUnprovable is invalid because it violates restrictions on orders on propositions. Nonexistence of the proposition I'mUnprovable motivated search for a proof of inferential undecidability of foundations.The Church/Turing Thesis is that a nondeterministic Turing Machine can perform any computation. The Thesis has been the overwhelming orthodoxy since the 1936. However, the Thesis is false because there are digital computations that cannot be performed by a nondeterministic Turing machine.Limitations of Church/Turing computation motivated development of ActorsTheory. The new theory characterized digital computation up to a unique isomorphism. In ActorsTheory, there is no procedure that can decide if an expression halts (“computational undecidability”). Inferential decidability implies that the halting problem is computationally decidable. Nonconstructive proof of inferential undecidability motivated search for a simple intuitive undecidable proposition.Settled consensus at least since Euclid was that “theorems of a theory can be algorithmically enumerated by starting with axioms and applying rules of inference using string substitutions.” However, [Church 1934] showed that a foundational theory cannot provably enumerate its own theorems. [Church 1934] concluded that foundations of mathematics are inconsistent because of the long established consensus. However, a very different conclusion results from formalizing [Church 1934] in ActorsTheory. Instead of being inconsistent, the theory is inferentially undecidable because the proposition StringTheoremsOfAnOrderAreEnumerable is neither provable nor disprovable. The proposition StringTheoremsOfAnOrderAreEnumerable is unprovable by the argument in [Church 1934]. Also, StringTheoremsOfAnOrderAreEnumerable is not disprovable because it is true in the unique up to isomporphism mathematical model of ActorsTheory. Consequently, ActorsTheory is inferentially incomplete because StringTheoremsOfAnOrderAreEnumerable is true but unprovable.Richard Dedekind vanquished monsters from the theory natural numbers by axiomatizing them up to a unique isomorphism using a higher-order induction axiom [Dedekind 1888]. Dedekind‘s results have been generalized in the theory Ordinals, which axiomatizes the ordinals up to a unique isomorphism. The theory Ordinals is much more powerful than the standard set theory ZFC. Axioms of ZFC are in effect axioms of the theory Ordinals. Furthermore, the type Ordinal is larger than any set in ZFC because no set can be placed in one-to-one correspondence with instances of Ordinal. Similarly, ActorsTheory vanquished monsters from digital computation. The theory axiomatizes digital computation up to a unique isomorphism using the Actor Event Induction axiom.Effective discourse between computer systems and humans (as well as between computer systems) is crucial for the development and deployment of Universal Intelligent Systems (UIS), which have pervasively inconsistent information.Creating and deploying Universal Intelligent Systems in this decade will pose huge new challenges for the foundations of Computer Science. Important developments in foundations include the following:• A Great Power strategic cyberattack can take down a country’s economy and keep it down for an indefinite period using reiterated cyberattacks.• Intelligent Systemso An Intelligent System is characterized by integration of large amounts of empirical information in close human interaction using multitudinous Intelligent Applications that are educable, adaptive-in-real-time, and self-informative.o Intelligent Systems will enable new kinds of cyberattacks.o Following rules outlined in this article can thwart many common cyberattacks. Also, attacks can be detected by attempted rule violations. o Development and universal deployment of Intelligent Systems in this decade will pose huge new challenges for resilience against cyberattacks.• ActorsTheoryo can implement important practical applications that are hundreds of times faster than any parallel lambda calculus expression.o has no nonlocal states because it is event-based.o characterizes digital computation up to a unique isomorphism thereby removing all ambiguity.o is effective because proof checking is computationally decidable even though theorems are not computationally enumerable thereby reinforcing [Church 1934] that the theorems of theories that axiomatize computation and provability cannot be computationally enumerable.o has orders only on propositional types thereby enabling usual mathematical practice using properly-typed recursive definitions using all types. Orders block construction of propositions that lead to inconsistency from [Russell 1908], [Gödel 1931], [Church 1934], etc. o allows all practically important fixed-point constructions of classical mathematics.• [Gödel 1931] seemed to have settled the issue of inferential undecidablity in the positive using the proposition I'mUnprovable. However, [Wittgenstein 1937] soon proved that existence of I'mUnprovable enables a cyberattack.• [Gödel 1931] failed to prove inferential incompleteness of foundations because construction of I'mUnprovable violated restrictions on orders of propositions.• Nonexistence of I'mUnprovable motivated search for a proof of inferential undecidability of foundations.• Nonconstructive proof of inferential undecidability using computational undecidability of the halting problem motivated search for a simple intuitive undecidable proposition.• Settled consensus at least since Euclid was that “theorems of a theory can be algorithmically enumerated by starting with axioms and applying rules of inference using string substitutions.” [Church 1934] concluded that foundations are inconsistent because the long established consensus enables a cyberattack.• ActorsTheory is inferentially undecidable because the proposition StringTheoremsOfAnOrderAreEnumerable is neither provable nor disprovable. Consequently, Hilbert's thesis that that every proposition is either provable or disprovable [Hilbert 1930] enables a cyberattack• ActorsTheory is inferentially incomplete because StringTheoremsOfAnOrderAreEnumerable is true in the unique up to isomorphism mathematical model of ActorsTheory but unprovable.• Undecidability and inferential incompleteness are not problems in practice.