Please tell me what is recursively enumerable in Logic and in Type Theory # ChatGPT Answer In logic and type theory, a set (or property, or predicate) is called recursively enumerable if there is an algorithm that will list out its members. That is, given enough time, this algorithm will enumerate all and only the elements of the set. The concept of recursive enumerability is important in areas such as computability theory and formal languages. It helps to delineate the boundary between what can be definitely known or done using a computer program and what cannot. For example, the set of all prime numbers is recursively enumerable because we can write a program that lists out all primes one by one. But the set of all true statements in arithmetic (also known as Gödel's set) is not recursively enumerable; there's no program that can list out all true arithmetic statements one by one. This concept also helps to understand undecidability problems in logic and computer science. A decision problem might be undecidable because although we can enumerate all potential solutions, there's no definite method to determine which ones are correct within a finite amount of steps. # Bard Answer **In type theory, recursively enumerable (RE) sets are collections of elements that can be systematically generated or enumerated by a Turing machine. Here's a breakdown of the concept:** **1. Turing Machines and Enumeration:** - A Turing machine is a theoretical model of computation that can simulate any algorithm or computer program. - It has a tape for storing data, a head for reading and writing, and a set of rules for manipulating symbols on the tape. - A set is RE if there exists a Turing machine that can list all of its elements, one by one, in some order. This means the machine can produce an infinite output stream containing every element of the set, even if it runs forever. **2. Type Theory and RE Sets:** - Type theory is a formal system for representing and reasoning about the structure of programs and mathematical proofs. - It uses types to classify expressions and ensure their consistency. - RE sets play a crucial role in type theory's ability to express computational properties and define logical structures. **3. Key Relationships and Applications:** - **Decidability:** A set is decidable if there exists a Turing machine that can determine whether any given element belongs to the set. All decidable sets are RE, but not all RE sets are decidable. - **Type checking:** RE sets are used in type checkers to determine whether a program is well-typed and free of type errors. - **Proof assistants:** RE sets are employed in proof assistants to automate theorem proving and ensure the correctness of logical arguments. - **Programming language design:** The concept of RE sets informs the design of programming languages, influencing features like type inference and language expressivity. **4. Examples of RE Sets:** - The set of all natural numbers (ℕ) is RE. - The set of all valid programs in a given programming language is RE. - The set of all provable theorems in a given logical system is RE. **5. Limitations and Beyond:** - While RE sets capture a wide range of computational phenomena, some sets are not RE, such as the set of all Turing machines that halt on a given input. - Research in type theory and computability theory explores extensions and alternatives to RE sets to address these limitations and capture even richer computational structures. **In summary, RE sets in type theory provide a powerful foundation for reasoning about computation, type systems, and logical reasoning. Their interplay with Turing machines and their applications in programming languages and proof assistants demonstrate their fundamental importance in theoretical computer science.** # References ```dataview Table title as Title, authors as Authors where contains(subject, "undefined") ```