Show that every finite subset of ℕ is primitive recursive. |...

Name: Problem 1, Chapter 5: Mathematical Logic: A Course with Exercises Part II: Recursion Theory, Godel's Theorems, Set Theory, Model Theory
Uploaded: 2019-10-10T13:08:03-08:00
Duration: 2 min 35 s
Channel: Angelo Rendina
Description: Show that every finite subset of ℕ is primitive recursive.

Key Concepts

Characteristic Functions

A characteristic function of a set maps each element to 1 if it is a member of the set and to 0 otherwise. For finite subsets of natural numbers, these functions can be defined by explicitly checking each possible member. Because the set is finite, the comparison and decision process can be expressed using primitive recursive operations.

Primitive Recursive Functions

Primitive recursive functions are a class of functions defined on the natural numbers that can be constructed from basic functions (like the zero, successor, and projection functions) using operations such as composition and primitive recursion. They are computable, always terminate on input, and form an important subclass of total computable functions.

Finite Sets

A finite set is a set with a limited number of elements. In the context of natural numbers, any finite subset can be handled explicitly by listing its elements. The limited size of the set allows algorithms to directly check membership by comparing the input number against each element in the set.

Closure Properties of Primitive Recursive Functions

Primitive recursive functions are closed under composition and primitive recursion, meaning that combining two or more such functions using these operations results in another primitive recursive function. This property is crucial in constructing decision procedures, such as the characteristic functions for finite sets, ensuring that these constructions remain within the primitive recursive framework.

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