Appendix B: Key Theorems Index

The Mathematics Ontology Bible · Version 1.0

Theorem	Part	Statement (compressed)
Gödel Completeness	I	⊢ ↔ ⊨ for FOL
Gödel Incompleteness I	X	Consistent sufficiently-expressive systems are incomplete
Gödel Incompleteness II	X	Such systems cannot prove their own consistency
Church-Turing Undecidability	IX	FOL validity is undecidable
Halting Problem	IX	No algorithm decides whether any TM halts
Cantor's Theorem	II
Continuum Hypothesis	II	Independent of ZFC
Yoneda Lemma	III	Objects determined by their relationships
Fundamental Theorem of Algebra	IV	ℂ is algebraically closed
Frobenius Theorem	IV	Only real division algebras are ℝ, ℂ, ℍ, 𝕆
Abel-Ruffini	V	No general quintic formula by radicals
Fundamental Theorem of Galois Theory	V	Subfields ↔ subgroups (order-reversing)
Spectral Theorem	VII	Self-adjoint operators diagonalizable
Dominated Convergence	VII	Limit and integral interchange under domination
Central Limit Theorem	VII	Sums of i.i.d. converge to Gaussian
Brouwer Fixed Point	VI	Continuous self-map of n-disk has fixed point
Poincaré Conjecture	VI	Simply-connected closed 3-manifold ≅ S³ (Perelman)
Four Color Theorem	VIII	Every planar graph is 4-colorable
Fermat's Last Theorem	VIII	No xⁿ+yⁿ=zⁿ for n≥3, x,y,z>0 (Wiles)
P vs. NP	IX	Open: Is P = NP?
Riemann Hypothesis	VII	Non-trivial ζ zeros on Re(s)=1/2 — Open
Noether's Theorem	XI	Symmetry ↔ conserved quantity
Gauss-Bonnet	VI	∫K dA = 2πχ(M)
Shannon Capacity	IX	Reliable communication rate = channel capacity
Cook's Theorem	IX	SAT is NP-complete