A bit of everything

# Axioms

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.

## Propositional logic

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

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.

## 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