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 $(a · b) · c = a · (b · c)$, identity $a · e = e · a = a$, inverses.

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

• Terms use variables, e, f.

• Atomic formulas: $(t1 = t2)$.

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 $\forall a \forall b \forall c ((a · b) · c = a · (b · c))$.

• 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 $P × P$.