Comprehensive taxonomy of formal mathematical vocabulary, notation systems, and foundational axioms. Covers mathematical logic symbols (\(\forall, \exists, \in, \subset, \implies, \iff\)), foundational set theories (ZFC axioms: Extensionality, Union, Power Set, Infinity, Choice), algebraic hierarchies (Magma \(\to\) Semigroup \(\to\) Monoid \(\to\) Group \(\to\) Ring \(\to\) Field), and standard topological properties (Compactness, Connectedness, Hausdorff separation).