Admissible Model

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An Admissible Model is a formal system of logic or mathematics that has been shown to be consistent and complete, meaning that it is possible to derive any valid formula or statement using the Axioms and Rules of the system. In other words, an Admissible Model satisfies all its Axioms.

History

The concept of an Admissible Model dates back to the early days of formal logic, where it was first introduced by Léon Pierre Émile du Châtelet in his 1829 book “A Treatise on Man and Society”. However, the modern definition of an Admissible Model as we know it today was formalized by Kurt Gödel in his 1931 paper “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”.

Definition

An Admissible Model is a formal system M = (A, P) that consists of:

• A set A of atomic formulas
• A set of predicate symbols P over A
• A set of Axioms A0, …, An of the form ∀x (φ(x) → ψ(x)) for all x in A
• A set of Rules R1, …, Rin of the form φ ∪ ψ if r ∩ ψ is ⊥

The model M satisfies the following properties:

• Consistency: For any formula φ, there exists a proof of φ in M that ends with a derivation of ⊥.
• Completeness: For every formula φ, either there exists a proof of φ in M or there exists a formula ψ such that there is no proof of φ in M.

Properties

Admissible Models have several important properties:

• Consistency: Admissible Models are consistent because any attempt to derive ⊥ will lead to a contradiction.
• Completeness: Admissible Models are complete because every valid formula can be proved or disproved using the Axioms and Rules of the model.
• Determinacy: Admissible Models are determined by their Axioms and Rules, meaning that there is only one possible interpretation of the model.

Examples

• Propositional Logic: A Propositional Logic system with a single axiom, where the only formula is ∧ (and), has an Admissible Model that consists of all atomic formulas.
• First-Order Logic: A First-Order Logic system with Axioms for basic logical connectives (e.g., ¬, →) and Rules such as conjunction introduction (i.e., if a, b are true then a ∩ b is true), also has an Admissible Model.

Applications

Admissible Models have a wide range of applications in mathematics and computer science:

• Proof Theory: Admissible Models provide a foundation for Proof Theory, which is the study of Formal Systems that can be used to prove statements.
• Model Checking: Admissible Models can be used to verify the correctness of software programs by simulating the execution of these programs using an Admissible Model.
• Verification: Admissible Models are often used in Verification tools such as Coq and Isabelle, which use them to check the correctness of mathematical proofs.

Conclusion

In conclusion, an Admissible Model is a formal system that has been shown to be consistent and complete, making it a fundamental concept in mathematics and computer science. Its properties provide a foundation for Proof Theory, Model Checking, and Verification, among other applications.