The arithmetic hierarchy is a term used in computer science and mathematics to describe a specific type of hierarchy related to the complexity of computational problems. Some common synonyms for the arithmetic hierarchy include the PH, polynomial hierarchy, and the Levy hierarchy. The PH refers to the polynomial hierarchy and is used to describe the complexity of decision problems that can be solved using polynomial time algorithms. The Levy hierarchy, on the other hand, is used to describe the complexity of problems that can be solved using only countable sequences of quantifiers. These synonyms are often used interchangeably depending on the specific context and field of study.