C0mmie said:
A set of rules for deriving true statements.
And how do you know if a statement is true? You need some other things too. Here's a quick outline.
You start with a language that contains a set of
symbols. You string the symbols together to get a set of
strings. You select some of the strings to get a set of
formulas.
You define a
valuation that tells you whether each formula is true or false. If a formula is true under every valuation (i.e. if it is always true), that formula is called a
tautology.
You then define a
calculus which consists of a set of
axioms and a set of
inference rules. If a formula can be derived from the calculus, that formula is called a
theorem. Now, soundness and completeness are properties of calculi. A calculus is
sound iff, for any formula F, if F is a theorem, then F is a tautology. A calculus is
complete iff, for any formula F, if F is a tautology, then F is a theorem.
Everything above, minus the valuation and tautologies, is called a
theory (or system). If every axiom or a set of rules to determine which formulas are axioms is given, then the theory is called an
axiomatic theory. If a theory has finitely many axioms or there can be given a finite set of rules to determine which formulas are axioms (i.e. an algorithm), then the theory is called
axiomatizable or
finitely axiomatizable.
Make sense? Do you have a statement of Gödel's Completeness or Incompleteness Theorems around? Edit: If not, you can search PF; They've been discussed many times here. Hurkyl and matt grime are reliable sources.
