Arithmetic hierarchy

Imagine you have a bunch of math problems, like adding numbers or multiplying them. The arithmetic hierarchy is like a list that tells you how complex these problems can get.

At the very bottom of the list are simple addition and subtraction problems, which are called "computable." We can solve these problems using a calculator or by counting with our fingers.

The next level up includes problems that are a bit more complicated, like finding the greatest common divisor of two numbers. These problems are called "arithmetical" and cannot be solved by a simple calculator.

As we move up the hierarchy, the problems become even more complex, involving more advanced concepts like sets, functions, and logical operations. These problems are called "analytical" and are much harder to solve.

At the very top of the hierarchy are problems that are so complex that they cannot be solved at all. These are called "noncomputable" problems, and they include things like the halting problem, which asks whether a computer program will ever finish running or not.

So, the arithmetic hierarchy is basically a way of organizing different levels of math problems based on how complex they are and how difficult they are to solve.