Hyperoperation - History

History

One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory of commutative hyperoperations (see below). About 12 years later, Wilhelm Ackermann defined the function which somewhat resembles the hyperoperation sequence.

In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, hexation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, 6, etc.). As a three-argument function, e.g., the hyperoperation sequence as a whole is seen to be a version of the original Ackermann function — recursive but not primitive recursive — as modified by Goodstein to incorporate the primitive successor function together with the other three basic operations of arithmetic (addition, multiplication, exponentiation), and to make a more seamless extension of these beyond exponentiation.

The original three-argument Ackermann function uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First, defines a sequence of operations starting from addition (n = 0) rather than the successor function, then multiplication (n = 1), exponentiation (n = 2), etc. Secondly, the initial conditions for result in, thus differing from the hyperoperations beyond exponentiation. The significance of the b + 1 in the previous expression is that =, where b counts the number of operators (exponentiations), rather than counting the number of operands ("a"s) as does the b in, and so on for the higher-level operations. (See the Ackermann function article for details.)