Mathematical Knowledge Brainstorming

This is a brainstorm.

=Category Tree=
 * Category:Lexeme The elements of the mathematical lexicon.
 * Category:Occurrence A mathematical object or example of a statement.
 * Category:Statement Statements like "The order of a subgroup divides the order of the group."
 * Category:Cluster

=Semantic Connection=


 * 1) Lexemes, Statements, Clusters, and Occurrences Property:has title a string.
 * 2) A lexeme Property:has lexical type noun, verb, adjective.
 * 3) A statement Property:has statement type theorem, corollary, lemma,axiom.
 * 4) A cluster Property:has cluster type axiomatization, theory, course.
 * 5) A lexeme/statement Property:semantically relies on a lexeme.
 * 6) A lexeme/statement Property:deductively relies on a statement.
 * 7) A cluster Property:contains a cluster/lexeme/statement.
 * 8) A statement Property:refines a statement.
 * 9) A lexeme Property:specializes a lexeme.
 * 10) A lexeme/statement Property:Has context a cluster.
 * 11) An occurrence Property:realizes a lexeme/statement.

=Set Theory= Everything is built up from set theory.

All definitions should be traceable back to the first-order language of set theory (LaST)

All constructions should be traceable back to the ZF-axioms (possibly ZFC).

= Page structure = One page per term defined.
 * 1) Terms semantically linked to lexemes ("normal subgroup" linked to normal, subgroup).
 * 2) Terms linked to all definitions, and all equivalence theorems.
 * 3) Terms linked to all existence proof theorems.
 * 4) Terms linked to examples (specifically constructed terms).
 * 5) Page has an informal paragraph description as well.

=Theorems=

The theorem types are:
 * 1) Existence/uniqueness theorems: A (something defined) exists (and is unique).
 * 2) Equivalence theorems: Given (some conditions).  The following conditions are equivalent:  (a list of conclusions).
 * 3) Hypothesis-conclusion theorems: Given (some conditions).  Then (some conclusion)
 * 4) Initial theorems: ZF(C).

Semantic connections:
 * 1) Theorems deductively require other theorems. All deduction goes back to ZF(C).
 * 2) Existence/uniqueness theorems prove existence/uniqueness of (something defined).