Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

History [of concept of mathematics] [History of] models of mathematics Axiom systems Basic logic [and axioms] Predicate logic [Axioms for] arithmetic Algebraic axioms Groups [and axioms] Semigroups [and axioms] Fields [and axioms] Rings [and axioms] Other algebraic systems Real algebra [and axioms] [Axioms for] geometry Category theory Set theory [and axioms] General topology [and axioms] [Axioms for] real analysis Axiom systems for programs Implementation [of proof example] Proof structures Substitution strategies [in proofs] One-way transformations [as axioms] Axiom schemas Reducing axiom [system] details [Mathematical] proofs in practice Properties [of example multiway systems] Nand tautologies [Methods for] proof searching Automated theorem proving Truth and falsity [in formal systems] Gödel's Theorem Properties [of example multiway systems] Essential incompleteness [in axiom systems] [Universality of] predicate logic [Universality of] algebraic axioms [Universality of] set theory Universal Diophantine equation Hilbert's Tenth Problem Polynomial value sets Statements in Peano arithmetic Transfinite numbers Growth rates [of functions] [Examples of] unprovable statements Encodings of arithmetic [by different operations] [The concept of] infinity Diophantine equations Properties [of Diophantine equations] Large solutions [to Diophantine equations] Nearby powers [and integer equations] Unsolved problems [in number theory] Fermat's Last Theorem More powerful axioms [for mathematics] Higher-order logics Truth and incompleteness Generalization in mathematics Cellular automaton axioms [Theorems about] practical programs Rules [for multiway systems examples] Consistency [in axiom systems] Properties [of example multiway systems] Non-standard arithmetic [Unprovable statements in] reduced arithmetic Generators and relations [and axiom systems] Comparison to multiway systems Operator systems [History of] truth tables Proofs of axiom systems Junctional calculus Equivalential calculus Implicational calculus Operators on sets Implementation [of operators from axioms] Properties [of operators from axioms] Algebraic systems [and operator systems] Symbolic systems [and operator systems] Groups and semigroups [and operator systems] Forcing of operators [by axiom systems] Model theory Pure equational logic Multiway systems [and operator systems] Logic in languages Properties [of logical primitives] Notations [for logical primitives] Universal logical functions Searching for logic [axioms] Two-operator logic [axioms] History [of logic axioms] Theorem distributions [in standard mathematics] Multivalued logic Proof lengths in logic Nand theorems Finite axiomatizability Empirical metamathematics Speedups in other systems Character of mathematics Invention versus discovery in mathematics Ordering of [mathematical] constructs Mathematics and the brain Frameworks [in mathematics]

From Stephen Wolfram: A New Kind of Science [citation]