**Derivation of A Theorem**

The notion of a theorem is very closely connected to its formal proof (also called a "derivation"). To illustrate how derivations are done, we will work in a very simplified formal system. Let us call ours Its alphabet consists only of two symbols { **A**, **B** } and its formation rule for formulas is:

- Any string of symbols of which is at least 3 symbols long, and which is not infinitely long, is a formula. Nothing else is a formula.

The single axiom of is:

**ABBA**

The only rule of inference (transformation rule) for is:

- Any occurrence of "
**A**" in a theorem may be replaced by an occurrence of the string "**AB**" and the result is a theorem.

Theorems in are defined as those formulae which have a derivation ending with that formula. For example

**ABBA**(Given as axiom)**ABBBA**(by applying the transformation rule)**ABBBAB**(by applying the transformation rule)

is a derivation. Therefore "**ABBBAB**" is a theorem of The notion of truth (or falsity) cannot be applied to the formula "**ABBBAB**" until an interpretation is given to its symbols. Thus in this example, the formula does not yet represent a proposition, but is merely an empty abstraction.

Two metatheorems of are:

- Every theorem begins with "
**A**". - Every theorem has exactly two "
**A**"s.

Read more about this topic: Theorem, Formalized Account of Theorems

### Famous quotes containing the word theorem:

“To insure the adoration of a *theorem* for any length of time, faith is not enough, a police force is needed as well.”

—Albert Camus (1913–1960)