A maximal consistent set is a collection of logical formulas that is both consistent and cannot be extended further without…
Gödel's incompleteness theorems reveal fundamental limits of formal systems. They demonstrate that any consistent system powerful enough for arithmetic will…
The property of a logical or mathematical system where not all true statements can be proven within the system's own…
Gödel's second incompleteness theorem states that no consistent formal system strong enough to include basic arithmetic can prove its own…
A theory is finitely axiomatizable if it can be completely defined by a finite collection of fundamental statements or axioms.…
A fundamental procedure in proof theory that systematically removes 'cuts' from a proof. This process simplifies proofs and demonstrates that…
A conservative extension adds new axioms or rules to a theory without altering the truth of existing statements. This ensures…