Propositions and Axiomatic Systems

Mathematical reasoning starts with statements whose meaning is precise enough to be assessed. The source chapter begins with propositions and then uses them to introduce the roles played by axioms, definitions, and theorems.

DefinitionProposition

A proposition is a declarative sentence that is either true or false. The sentence must be sufficiently precise for its truth value to be determined.

Questions, commands, and subjective opinions are not propositions in this sense. A proposition may be false and still be a proposition. For example, “every prime number is even” is false, but it is a declarative statement with a definite truth value.

The components of an axiomatic system

An axiomatic system consists of a formal language, axioms, definitions, and rules of inference. Within such a system, the following distinctions are useful:

• An axiom is a statement adopted as a starting assumption.
• A definition fixes the meaning of a term. It is not normally classified as true or false.
• A theorem is a proposition derived from the axioms, definitions, and earlier results.
• A lemma is a theorem introduced mainly as a tool for proving another result.
• A corollary follows readily from a theorem or lemma.

The classification depends on the formal system. A statement that is taken as an axiom in one development may be proved as a theorem in another.

ExampleA small algebraic derivation

For integers $m$ and $n$, the exponent law gives

$$a^m a^n = a^{m+n}.$$

If addition is commutative, then $m+n=n+m$, so

$$a^m a^n = a^{m+n}=a^{n+m}=a^n a^m.$$

The derivation depends on the exponent law and commutativity. It is not an argument from intuition alone.

Consistency, independence, and completeness

These properties describe different aspects of a formal system.

DefinitionConsistency

A formal system is consistent if it does not prove both a proposition and its negation. In symbols, there is no proposition $P$ for which both $P$ and $\neg P$ are derivable.

DefinitionIndependence

An axiom is independent of the remaining axioms if it cannot be derived from them. Removing an independent axiom can therefore produce a genuinely weaker theory.

DefinitionCompleteness

A formal system is complete when every sentence in its language is decided by the system, meaning that the sentence or its negation is derivable. This is a property of a formal theory, not a claim that the theory contains every mathematical truth.

The fifth postulate of Euclidean geometry provides a classical example of independence. Replacing it with a different parallel axiom leads to non-Euclidean geometries. This does not show that mathematics has no truth; it shows that conclusions are relative to the assumptions and rules of the theory in which they are proved.

That distinction is important: truth in a mathematical model, derivability in a formal system, and the choice of axioms are related but not identical notions.