8. Theorems and proofs

Theorems and proofs

Theorem

A theorem is when a proposition can be shown to be true and is important enough.

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A proof of a theorem, $t$, is a sequence of propositions ending with $t$:

$$\begin{array}{rlrl}& \underline{\begin{array}{l}{p}_1\\ p_2\\ ..\\ p_n\end{array}\}}& & n\text{ premises}\\ & q& & \end{array}$$

Such that each and every of the premises is:

• Either an axiom.
• Or can be obtained from earlier premises in the sequence by a logically correct argument.

It is said that theorem $t$ follows from the axioms.

How are theorems stated

Many theorems assert that a property holds for all entities in a domain. The precise statement must include a universal quantifier, standard convention in mathematics is to omit it.

• The statement “if $x>5$, where $x$ is a positive integer, then $x^2>30$”
• Means that “for all positive integers $x$, if $x>5$, then $x^2>30$”