Notes on Mathematical Language and First-Order Logic

The Language of Mathematics

  • Mathematics employs formal language for precise, unambiguous proofs.

  • Logical connectives build complex expressions from simpler ones.

  • Formulas are constructed recursively, using parentheses to ensure clarity.

  • Propositional logic simplifies mathematical language.

Quantifiers and Variables

  • Mathematical language uses quantifiers ∀\forall (for all) and ∃\exists (there exists).

  • Variables are used with quantifiers in mathematical domains.

  • Primitive statements serve as foundational axioms.

Group Theory Example

  • A group is set G with operation ⋅· and element e.

  • Group axioms: associativity (a⋅b)⋅c=a⋅(b⋅c)(a · b) · c = a · (b · c), identity a⋅e=e⋅a=aa · e = e · a = a, inverses.

  • Function notation expresses axioms: f(a,b)=a⋅bf(a, b) = a · b.

  • Terms use variables, e, f.

  • Atomic formulas: (t1=t2)(t1 = t2).

Formulas with Atomic Formulas and Connectives

  • Formulas are built using connectives and quantifiers for complex relationships.

  • Group axioms are expressed with quantifiers and connectives, such as ∀a∀b∀c((a⋅b)⋅c=a⋅(b⋅c))\forall a \forall b \forall c ((a · b) · c = a · (b · c)).

  • Syntactically correct formulas can be unintuitive.

Commutative Ring Theory

  • Commutative ring theory uses addition and multiplication symbols with 0 and 1.

  • Axioms for rings can be written straightforwardly.

Theory of Partial Orderings

  • Partial ordering is defined on set P with subset << of P×PP × P.