Grzegorczyk hierarchy

The Grzegorczyk hierarchy (pronounced: [ɡʐɛˈɡɔrt͡ʂɨk]), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory (Wagner and Wechsung 1986:43). Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower level of the hierarchy grow slower than functions in the higher levels.

First we introduce an infinite set of functions, denoted E_i for some natural number i. We define ${\displaystyle E_{0}(x,y)=x+y}$ and ${\displaystyle E_{1}(x)=x^{2}+2}$. I.e., E₀ is the addition function, and E₁ is a unary function which squares its argument and adds two. Then, for each n greater than 1, we define ${\displaystyle E_{n}(x)=E_{n-1}^{x}(2)}$, i.e. the x-th iterate of ${\displaystyle E_{n-1}}$ evaluated at 2.

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