Reverse Mathematics and Countable Algebraic Systems

This thesis is a contribution to the foundation of mathematics, especially to the Reverse Mathematics program, whose core question is “What are the appropriate axioms to prove each mathematical theorem?” The main targets of this thesis are countable algebraic systems and its theories in second order arithmetic. This thesis establishes that several theorems and the existence of objects of countable algebraic systems are equivalent to specific set existence axioms— arithmetical comprehension or weak König’s lemma. These equivalences are proven within RCA0, a subsystem of arithmetic mainly consisting of recursive comprehension and Σ1 induction. Weak König’s lemma is a non-constructive (however, strictly weaker than arithmetical comprehension) set existence axiom which asserts that every infinite 0-1 tree has a path. In Chapter 3, we develop countable order theory in second order arithmetic. We prove within RCA0 that a countable version of the Abian-Brown least fixed point theorem, Davis’ converse, Markowsky’s converse, and arithmetical comprehension are pairwise equivalent. We also show that a countable version of the Knaster-Tarski fixed point theorem, the Tarski-Kantorovitch fixed point theorem, and the Bourbaki-Witt fixed point theorem are provable within RCA0. In Chapter 4, we develop countable semigroup theory in second order arithmetic. We prove within RCA0 that Isbell’s zig-zag theorem for countable monoids is equivalent to weak König’s lemma, and that the existence of dominions is equivalent to arithmetical comprehension. We also show that the Rees theorem for countable semigroups is implied via arithmetical comprehension. In Chapter 5, we develop countable group theory in second order arithmetic. We prove within RCA0 that the existence of essential closures (as known as neat hulls), normalizers, and abelianizers (as known as derived subgroups or commutator groups) are equivalent to arithmetical comprehension. We also show that characterizations of normalizers and ablianizer are equivalent to weak König’s lemma. In Chapter 6, we develop countable commutative ring theory or ideal theory in second order arithmetic. We show within RCA0 that the existence of the sum, the product, the quotient, and the radical of two ideals is equivalent to arithmetical comprehension. We also show that the Lasker-Noether primary ideal decomposition theorem for countable commutative rings is implied via arithmetical comprehension.