Mathematical reasoning, logic, and set theory

In order to communicate mathematical ideas effectively, formal languages have been developed within which logic, i.e. deductive (mathematical) reasoning, can proceed. Propositions are statements that can be either true $\mathrm{\top }$ or false $\mathrm{\perp }$. Axiomatic systems begin with statements (axioms) assumed true. Theorems are proven by deduction. In many forms of logic, like propositional calculus (Wikipedia 2019b), compound propositions are constructed via logical connectives like “and” and “or” of atomic propositions (see section 2.2). In others, like first-order logic (Wikipedia 2019a), there are also logical quantifiers like “for every” and “there exists.”

The mathematical objects and operations about which most propositions are made are expressed in terms of set theory, which was introduced in section 1.2 and will be expanded upon in section 2.1. We can say that mathematical reasoning is comprised of mathematical objects and operations expressed in set theory and logic allows us to reason therewith.