Axiom:Peano's Axioms/Also defined as
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Peano's axioms: Also defined as
Some treatments of Peano's axioms define the non-successor element (or primal element) to be $1$ and not $0$.
The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.
Sources
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 4$: The natural numbers
- 1964: J. Hunter: Number Theory ... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers