As for every formal system, we require a handful of axioms as fundamental units. Here are the most common axioms and axiom schemes we may require.
The most fundamental and most common axiom system is that of propositional logic. It is composed of atomic propositions (usually noted by $p, q, r, \ldots$), and we can also construct a proposition from any other proposition using some logical operators. The most common basic ones are the negation $\neg$, leading to the proposition $\neg p$, the disjunction $\vee$, leading to the proposition $p \vee q$, the implication $\to$, leading to the proposition $p \to q$, and the conjunction, leading to the proposition $p \wedge q$.
Axioms and rules of inference
- The modus ponens : $$\vdash p, \vdash (p \to q) \Rightarrow \vdash q$$
- The principle of simplification : $$\vdash (p \to (q \to p))$$
- Frege's axiom : $$\vdash ((p \to (q \to r) \to ((p \to r) \to (p \to r)))$$
Predicate calculus grafts itself on top of propositional calculus. In addition to the previous set of propositions, we also have, for any proposition $p$, that $\forall x, p$ is also a proposition.
ZFC set theory
the real and complex numbers
It is common to define the properties of real and complex numbers independently of ZFC, even though it is possible