Exhibit a recursive set A ⊆ℕ^2 such that the set B={x : for all y ∈ℕ,(x, y) ∈A} is not recursi

step 1 Okay, so we want to prove that A, UB is an infinite set if A is an infinite set. Well, this is v

Exhibit a recursive set A ⊆ℕ^2 such that the set B={x : for...

Key Concepts

Recursive Sets

A set is said to be recursive (or decidable) if there exists an algorithm (or Turing machine) that can determine, in a finite amount of time, whether any given element belongs to the set. In computability theory, recursive sets are central because they represent problems with a complete solution method, where the yes/no decision can always be made effectively.

Recursively Enumerable Sets

A recursively enumerable (RE) set is one for which there is an algorithm that will list all the elements of the set, possibly without halting for non?elements. Although membership might not be decidable in finite time, if an element is indeed in an RE set, there is a computable procedure that will eventually confirm its membership. This concept is crucial for understanding problems that are semi-decidable but not necessarily decidable.

Arithmetical Hierarchy

The arithmetical hierarchy is a classification of decision problems based on the complexity of the logical formulas that define them, particularly the number and type of quantifiers used. Problems that involve a universal quantifier over natural numbers (as in ‘for all y’) often fall into a higher level of this hierarchy, which can mean that even if a related set is defined by a decidable relation, the resulting set may not be recursively enumerable.

Universal Quantification in Computability

Universal quantification in the definition of a set introduces a level of complexity because verifying that 'for all y some property holds' generally requires checking infinitely many cases. This increases the complexity of deciding membership and can lead to sets that, even if defined using components that are decidable, are themselves not recursively enumerable. This concept is crucial in understanding how simple constructions can elevate the complexity of a problem to a level beyond the reach of effective enumeration.

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