Automata Theory Questions Medium
In the field of mathematical logic, both Frege and extended Frege proof systems are formal systems used to derive logical proofs. However, there are some key differences between the two.
1. Scope of axioms: In a Frege proof system, the axioms are typically limited to the basic logical laws, such as the laws of propositional logic and predicate logic. On the other hand, an extended Frege proof system allows for additional axioms beyond the basic logical laws. These additional axioms can include specific mathematical principles or rules that are not part of the standard logical laws.
2. Expressive power: Due to the inclusion of additional axioms, the extended Frege proof system has a higher expressive power compared to the Frege proof system. This means that the extended Frege system can prove a wider range of statements and theorems than the Frege system alone.
3. Complexity: The extended Frege proof system is generally more complex than the Frege proof system. This complexity arises from the inclusion of additional axioms and the resulting increase in the number of rules and inference steps required to derive a proof.
4. Applications: The Frege proof system is often used as a foundation for formalizing logical reasoning and proving theorems in various branches of mathematics. It provides a solid basis for understanding the fundamental principles of logic. On the other hand, the extended Frege proof system finds applications in more specialized areas of mathematics and logic where the additional axioms are necessary to capture specific mathematical concepts or principles.
Overall, the main difference between a Frege and extended Frege proof system lies in the scope of axioms, expressive power, complexity, and applications. The extended Frege system extends the basic logical laws with additional axioms, resulting in a more powerful but also more complex proof system.