Algebraic axioms

Axioms like a ∘ (b ∘ c) (a ∘ b) ∘ c can be used in at least three ways. First, as equations which can be manipulated—like the axioms of basic logic—to establish whether expressions are equal. Second, as on page 773, as statements to be added to the axioms of predicate logic to yield results that hold for every possible system described by the axioms (say every possible semigroup). And third, as definitions of sets whose properties can be studied—and compared—using set theory. High-school algebra typically treats axioms as equations. More advanced algebra often uses predicate logic, but implicitly uses set theory whenever it addresses for example mappings between objects. Note that as discussed on page 1159 how one uses algebraic axioms can affect issues of universality and undecidability. (See also page 1169.)