Natural Number Multiplication is Unique

Theorem

Let $\N$ denote the natural numbers.

Let $M: \N \times \N \to \N$ be the mapping defined as the multiplication operation:

$\forall \tuple {x, y} \in \N \times \N: \map M {x, y} = \begin {cases} 0 & : y = 0 \\ \paren {\map M {x, r} } + r & : y = r + 1 \end {cases}$

where $+$ denotes natural number addition.

Then $M$ exists and is unique.

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 8$ Definition by finite recursion: Exercise $8.4 \ \text {(b)}$