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Notes
Chapter 12
Section 9
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Notes
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]