Hyperoperation in the context of Successor function

⭐ Core Definition: Hyperoperation

In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).

After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written using n − 2 arrows in Knuth's up-arrow notation.Each hyperoperation may be understood recursively in terms of the previous one by:

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👉 Hyperoperation in the context of Successor function

In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function.

Successor operations are also known as zeration in the context of a zeroth hyperoperation: H₀(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.

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Hyperoperation in the context of Successor function

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⭐ Core Definition: Hyperoperation

👉 Hyperoperation in the context of Successor function