# 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