Arithmetical Comprehension and Turing Jump Closure for $\omega$-Models Logic Generalized von Neumann Theorem for Arithmetic Progressions Combinatorics Induction on Derivation Height Logic Frame-Robinson-Thrall Hook-Length Formula Combinatorics Countability of Finite Subsets Discrete Mathematics Transitivity of Many-One Reducibility Logic Bogolyubov's Theorem for Finite Abelian Groups Combinatorics Regular Languages Are Closed Under Concatenation Discrete Mathematics Discrete Mathematics Area Combinatorics Subarea