Here's a breakdown:
* Formal System: A formal system consists of:
* Alphabet: A set of symbols.
* Syntax: Rules for forming well-formed formulas (WFFs) from the alphabet. These are the "sentences" of the system.
* Axioms: A set of initial WFFs assumed to be true.
* Inference Rules: Rules for deriving new WFFs (theorems) from existing ones.
* Gödel Completeness (for First-Order Logic): A first-order logical system is *Gödel complete* if every logically valid sentence (a sentence that is true in all interpretations) is provable within the system. In simpler terms: if a statement is true according to the system's rules of logic, then the system can actually prove that statement.
* Completeness in other contexts: The notion of completeness can be adapted to other areas of mathematics. For example, in a specific mathematical theory (like group theory or number theory), a set of axioms is considered complete if it allows the derivation of all true statements expressible within that theory. However, this often implies a limitation on what can be expressed within the theory itself.
The crucial point about completeness: Gödel's incompleteness theorems show that sufficiently complex formal systems (those capable of expressing basic arithmetic) cannot be both complete and consistent. Consistency means that the system doesn't prove both a statement and its negation. Therefore, any sufficiently powerful consistent formal system will inevitably contain true statements that cannot be proven within the system. This means there's no single, perfectly complete mathematical system encompassing all of mathematics. Each system will have its own scope and limitations regarding completeness.