Proof Theory encompasses the study of natural deduction, axiomatic systems, and formal languages, which are all fundamental concepts in the field.

Proof Theory aims to achieve all of these goals, as it seeks to understand the nature of proofs, their construction, and the properties of the systems in which they are formulated.

In Proof Theory, a formal language consists of a set of symbols and a set of rules that govern how these symbols can be combined to form well-formed formulas.

A deductive system in Proof Theory comprises a set of axioms, which are statements assumed to be true, and a set of inference rules, which allow for the derivation of new statements from existing ones.

Axioms serve multiple purposes in a deductive system. They provide a foundation for proofs, define the basic concepts of the system, and establish its fundamental properties.

Inference rules in a deductive system provide the means to derive new statements from existing ones. They define the logical steps that can be taken to reach new conclusions based on the given axioms and previously derived statements.

In Proof Theory, a proof is a sequence of statements, each of which is either an axiom or is derived from previous statements in the sequence using inference rules. It establishes the validity of a conclusion based on the given axioms.

In Proof Theory, a theorem is a statement that has been proven to be true within a deductive system, while a proof is the sequence of statements that demonstrates the validity of the theorem.

Soundness and completeness are crucial properties of deductive systems. Soundness guarantees that any statement derived as a theorem is true, while completeness ensures that any true statement can be proven within the system.

Gentzen's natural deduction system is a widely recognized natural deduction system that provides a foundation for studying the structure and validity of proofs.

Sequent calculus serves multiple purposes in Proof Theory. It offers a framework for analyzing the structure of proofs, helps establish the soundness and completeness of deductive systems, and facilitates the investigation of the properties of logical connectives.

Gödel's incompleteness theorems are considered fundamental results in Proof Theory. They demonstrate the limitations of formal systems and highlight the existence of statements that cannot be proven or disproven within those systems.

Gödel's incompleteness theorems have profound implications in Proof Theory. They demonstrate the inherent limitations of formal systems, provide a method for constructing undecidable statements, and establish the existence of true statements that cannot be proven within those systems.

Automated theorem proving, proof complexity, and intuitionistic logic are all active areas of research in Proof Theory. They explore the automation of proof construction, the computational complexity of proofs, and the study of non-classical logics, respectively.