Proof theory · Definition
Axiomatic system
Before you start you need to know
An axiomatic system organizes mathematical reasoning into three parts: a language, starting axioms and rules for deriving results.
Definition
An axiomatic system is a triple \(\mathcal{S}=(\mathcal{L},\mathcal{A},\mathcal{R})\) made of a formal language, axioms and rules of inference.
Every theorem of an axiomatic system has a finite derivation, so each result can be checked step by step without relying on intuition.
Take the axioms \(p\to(q\to p)\) and \(p\), with modus ponens. Then \(q\to p\) is a theorem: it follows from \(p\) and \(p\to(q\to p)\) in one step.
A typical mistake is to think an axiom is just an obvious truth that need not be stated. Inside the system only what is listed counts; an obvious-looking formula is not a theorem until it has a derivation.
Think about it
Which three minimal ingredients appear in an axiomatic system?
A language, axioms and rules of inference.
Exercises 11
- An axiomatic system includes...
- Axioms work as...
- A derivation inside an axiomatic system is...
- The rules of an axiomatic system indicate...
- An axiomatic system can be studied by asking whether it is...
- An axiomatic system starts from...
- An axiom is a statement that...
- The rules of inference indicate...
- Which of these statements about a formal system is correct?
- A proof inside a system depends on...
- Which of these formulas is a theorem of the system with axioms \(p\), \(p\to q\) and \(q\to r\), using modus ponens?
What comes next
References
- Frederic P. Schuller, Lectures on the Geometrical Anatomy of Theoretical Physics · Lecture 1, Introduction / Logic of Propositions and Predicates, YouTube, 2015.