Adequate pointclass

Last updated

In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive substitution, bounded universal and existential quantification and preimages by recursive functions. [1] [2]

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.

In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass.

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time and correctly decides whether the number belongs to the set or not.

Related Research Articles

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Theory of computation subfield of computer science

In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".

In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics. In type theory, every "term" has a "type" and operations are restricted to terms of a certain type.

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1 inclusive. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

Computability theory, also known as recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory.

Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Emil Leon Post American logician

Emil Leon Post was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.

Syntax (logic)

In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.

In computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.

Yiannis N. Moschovakis American logician

Yiannis Nicholas Moschovakis is a set theorist, descriptive set theorist, and recursion (computability) theorist, at UCLA.

In recursion theory a subset of the natural numbers is called a simple set if it is co-infinite and recursively enumerable, but every infinite subset of its complement fails to be enumerated recursively. Simple sets are examples of recursively enumerable sets that are not recursive.

Gerald Enoch Sacks is a logician who holds a joint appointment at Harvard University as a professor of mathematical logic and the Massachusetts Institute of Technology as a professor emeritus. His most important contributions have been in recursion theory. Named after him is Sacks forcing, a forcing notion based on perfect sets and the Sacks Density Theorem, which asserts that the partial order of the recursively enumerable Turing degrees is dense.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

In mathematics and computer science, the BIT predicate or Ackermann coding, sometimes written BIT(ij), is a predicate which tests whether the jth bit of the number i is 1, when i is written in binary.

In mathematical logic, a theory is complete if, for every formula in the theory's language, that formula or its negation is demonstrable. Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem.

In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within the lattice of the recursively enumerable sets.

In logic, mathematics and computer science, especially metalogic and computability theory, an effective method or effective procedure is a procedure for solving a problem from a specific class. An effective method is sometimes also called a mechanical method or procedure.

In proof theory, ordinal analysis assigns ordinals to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0.

In computability theory, recursively inseparable sets are pairs of sets of natural numbers that cannot be "separated" with a recursive set. These sets arise in the study of computability theory itself, particularly in relation to Π0
1
classes
. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.

References

  1. Moschovakis, Y. N. (1987), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, Elsevier, p. 158, ISBN   9780080963198 .
  2. Gabbay, Dov M.; Kanamori, Akihiro; Woods, John (2012), Sets and Extensions in the Twentieth Century, Handbook of the History of Logic, 6, Elsevier, p. 465, ISBN   9780080930664 .