![History [of concept of mathematics]](/nks/img/thumbnails/notes-12-9--history-of-concept-of-mathematics--textonly.png "History [of concept of mathematics]")
![[History of] models of mathematics](/nks/img/thumbnails/notes-12-9--history-of-models-of-mathematics--textonly.png "[History of] models of mathematics")
![Basic logic [and axioms]](/nks/img/thumbnails/notes-12-9--basic-logic-and-axioms--textonly.png "Basic logic [and axioms]")
![[Axioms for] arithmetic](/nks/img/thumbnails/notes-12-9--axioms-for-arithmetic--textonly.png "[Axioms for] arithmetic")
![Groups [and axioms]](/nks/img/thumbnails/notes-12-9--groups-and-axioms--textonly.png "Groups [and axioms]")
![Semigroups [and axioms]](/nks/img/thumbnails/notes-12-9--semigroups-and-axioms--textonly.png "Semigroups [and axioms]")
![Fields [and axioms]](/nks/img/thumbnails/notes-12-9--fields-and-axioms--textonly.png "Fields [and axioms]")
![Rings [and axioms]](/nks/img/thumbnails/notes-12-9--rings-and-axioms--textonly.png "Rings [and axioms]")
![Real algebra [and axioms]](/nks/img/thumbnails/notes-12-9--real-algebra-and-axioms--textonly.png "Real algebra [and axioms]")
![[Axioms for] geometry](/nks/img/thumbnails/notes-12-9--axioms-for-geometry--textonly.png "[Axioms for] geometry")
![Set theory [and axioms]](/nks/img/thumbnails/notes-12-9--set-theory-and-axioms--textonly.png "Set theory [and axioms]")
![General topology [and axioms]](/nks/img/thumbnails/notes-12-9--general-topology-and-axioms--textonly.png "General topology [and axioms]")
![[Axioms for] real analysis](/nks/img/thumbnails/notes-12-9--axioms-for-real-analysis--textonly.png "[Axioms for] real analysis")
![Implementation [of proof example]](/nks/img/thumbnails/notes-12-9--implementation-of-proof-example--textonly.png "Implementation [of proof example]")
![Substitution strategies [in proofs]](/nks/img/thumbnails/notes-12-9--substitution-strategies-in-proofs--textonly.png "Substitution strategies [in proofs]")
![One-way transformations [as axioms]](/nks/img/thumbnails/notes-12-9--one-way-transformations-as-axioms--textonly.png "One-way transformations [as axioms]")
![Reducing axiom [system] details](/nks/img/thumbnails/notes-12-9--reducing-axiom-system-details--textonly.png "Reducing axiom [system] details")
![[Mathematical] proofs in practice](/nks/img/thumbnails/notes-12-9--mathematical-proofs-in-practice--textonly.png "[Mathematical] proofs in practice")
![Properties [of example multiway systems]](/nks/img/thumbnails/notes-12-9--properties-of-example-multiway-systems-1--textonly.png "Properties [of example multiway systems]")
![[Methods for] proof searching](/nks/img/thumbnails/notes-12-9--methods-for-proof-searching--textonly.png "[Methods for] proof searching")
![Truth and falsity [in formal systems]](/nks/img/thumbnails/notes-12-9--truth-and-falsity-in-formal-systems--textonly.png "Truth and falsity [in formal systems]")
![Properties [of example multiway systems]](/nks/img/thumbnails/notes-12-9--properties-of-example-multiway-systems-2--textonly.png "Properties [of example multiway systems]")
![Essential incompleteness [in axiom systems]](/nks/img/thumbnails/notes-12-9--essential-incompleteness-in-axiom-systems--textonly.png "Essential incompleteness [in axiom systems]")
![[Universality of] predicate logic](/nks/img/thumbnails/notes-12-9--universality-of-predicate-logic--textonly.png "[Universality of] predicate logic")
![[Universality of] algebraic axioms](/nks/img/thumbnails/notes-12-9--universality-of-algebraic-axioms--textonly.png "[Universality of] algebraic axioms")
![[Universality of] set theory](/nks/img/thumbnails/notes-12-9--universality-of-set-theory--textonly.png "[Universality of] set theory")
![Growth rates [of functions]](/nks/img/thumbnails/notes-12-9--growth-rates-of-functions--textonly.png "Growth rates [of functions]")
![[Examples of] unprovable statements](/nks/img/thumbnails/notes-12-9--examples-of-unprovable-statements--textonly.png "[Examples of] unprovable statements")
![Encodings of arithmetic [by different operations]](/nks/img/thumbnails/notes-12-9--encodings-of-arithmetic-by-different-operations--textonly.png "Encodings of arithmetic [by different operations]")
![[The concept of] infinity](/nks/img/thumbnails/notes-12-9--the-concept-of-infinity--textonly.png "[The concept of] infinity")
![Properties [of Diophantine equations]](/nks/img/thumbnails/notes-12-9--properties-of-diophantine-equations--textonly.png "Properties [of Diophantine equations]")
![Large solutions [to Diophantine equations]](/nks/img/thumbnails/notes-12-9--large-solutions-to-diophantine-equations--textonly.png "Large solutions [to Diophantine equations]")
![Nearby powers [and integer equations]](/nks/img/thumbnails/notes-12-9--nearby-powers-and-integer-equations--textonly.png "Nearby powers [and integer equations]")
![Unsolved problems [in number theory]](/nks/img/thumbnails/notes-12-9--unsolved-problems-in-number-theory--textonly.png "Unsolved problems [in number theory]")
![More powerful axioms [for mathematics]](/nks/img/thumbnails/notes-12-9--more-powerful-axioms-for-mathematics--textonly.png "More powerful axioms [for mathematics]")
![[Theorems about] practical programs](/nks/img/thumbnails/notes-12-9--theorems-about-practical-programs--textonly.png "[Theorems about] practical programs")
![Rules [for multiway systems examples]](/nks/img/thumbnails/notes-12-9--rules-for-multiway-systems-examples--textonly.png "Rules [for multiway systems examples]")
![Consistency [in axiom systems]](/nks/img/thumbnails/notes-12-9--consistency-in-axiom-systems--textonly.png "Consistency [in axiom systems]")
![Properties [of example multiway systems]](/nks/img/thumbnails/notes-12-9--properties-of-example-multiway-systems-3--textonly.png "Properties [of example multiway systems]")
![[Unprovable statements in] reduced arithmetic](/nks/img/thumbnails/notes-12-9--unprovable-statements-in-reduced-arithmetic--textonly.png "[Unprovable statements in] reduced arithmetic")
![Generators and relations [and axiom systems]](/nks/img/thumbnails/notes-12-9--generators-and-relations-and-axiom-systems--textonly.png "Generators and relations [and axiom systems]")
![[History of] truth tables](/nks/img/thumbnails/notes-12-9--history-of-truth-tables--textonly.png "[History of] truth tables")
![Implementation [of operators from axioms]](/nks/img/thumbnails/notes-12-9--implementation-of-operators-from-axioms--textonly.png "Implementation [of operators from axioms]")
![Properties [of operators from axioms]](/nks/img/thumbnails/notes-12-9--properties-of-operators-from-axioms--textonly.png "Properties [of operators from axioms]")
![Algebraic systems [and operator systems]](/nks/img/thumbnails/notes-12-9--algebraic-systems-and-operator-systems--textonly.png "Algebraic systems [and operator systems]")
![Symbolic systems [and operator systems]](/nks/img/thumbnails/notes-12-9--symbolic-systems-and-operator-systems--textonly.png "Symbolic systems [and operator systems]")
![Groups and semigroups [and operator systems]](/nks/img/thumbnails/notes-12-9--groups-and-semigroups-and-operator-systems--textonly.png "Groups and semigroups [and operator systems]")
![Forcing of operators [by axiom systems]](/nks/img/thumbnails/notes-12-9--forcing-of-operators-by-axiom-systems--textonly.png "Forcing of operators [by axiom systems]")
![Multiway systems [and operator systems]](/nks/img/thumbnails/notes-12-9--multiway-systems-and-operator-systems--textonly.png "Multiway systems [and operator systems]")
![Properties [of logical primitives]](/nks/img/thumbnails/notes-12-9--properties-of-logical-primitives--textonly.png "Properties [of logical primitives]")
![Notations [for logical primitives]](/nks/img/thumbnails/notes-12-9--notations-for-logical-primitives--textonly.png "Notations [for logical primitives]")
![Searching for logic [axioms]](/nks/img/thumbnails/notes-12-9--searching-for-logic-axioms--textonly.png "Searching for logic [axioms]")
![Two-operator logic [axioms]](/nks/img/thumbnails/notes-12-9--two-operator-logic-axioms--textonly.png "Two-operator logic [axioms]")
![History [of logic axioms]](/nks/img/thumbnails/notes-12-9--history-of-logic-axioms--textonly.png "History [of logic axioms]")
![Theorem distributions [in standard mathematics]](/nks/img/thumbnails/notes-12-9--theorem-distributions-in-standard-mathematics--textonly.png "Theorem distributions [in standard mathematics]")
![Ordering of [mathematical] constructs](/nks/img/thumbnails/notes-12-9--ordering-of-mathematical-constructs--textonly.png "Ordering of [mathematical] constructs")
![Frameworks [in mathematics]](/nks/img/thumbnails/notes-12-9--frameworks-in-mathematics--textonly.png "Frameworks [in mathematics]")
History [of concept of mathematics]
[History of] models of mathematics
Implementation [of proof example]
Substitution strategies [in proofs]
One-way transformations [as axioms]
Reducing axiom [system] details
[Mathematical] proofs in practice
Properties [of example multiway systems]
Truth and falsity [in formal systems]
Properties [of example multiway systems]
Essential incompleteness [in axiom systems]
[Universality of] predicate logic
[Universality of] algebraic axioms
Universal Diophantine equation
Statements in Peano arithmetic
[Examples of] unprovable statements
Encodings of arithmetic [by different operations]
Properties [of Diophantine equations]
Large solutions [to Diophantine equations]
Nearby powers [and integer equations]
Unsolved problems [in number theory]
More powerful axioms [for mathematics]
[Theorems about] practical programs
Rules [for multiway systems examples]
Consistency [in axiom systems]
Properties [of example multiway systems]
[Unprovable statements in] reduced arithmetic
Generators and relations [and axiom systems]
Comparison to multiway systems
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]
Multiway systems [and operator systems]
Properties [of logical primitives]
Notations [for logical primitives]
Theorem distributions [in standard mathematics]
Invention versus discovery in mathematics