# Axiom:Peano's Axioms/Historical Note

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## Historical Note on Peano's Axioms

A set of axioms on the same topic as **Peano's axioms** was initially formulated by Richard Dedekind in $1888$.

Giuseppe Peano published them in $1889$ according to his own formulation, in a more precisely stated form than Dedekind's.

Bertrand Russell pointed out that while **Peano's axioms** give the key properties of the natural numbers, they do not actually define what the natural numbers actually are.

According to 1960: Paul R. Halmos: *Naive Set Theory*:

*[These] assertions ... are known as the Peano axioms; they used to be considered as the fountainhead of all mathematical knowledge.*

It is worth pointing out that the **Peano axioms** can be deduced to hold for the minimally inductive set as defined by the Axiom of Infinity from the Zermelo-Fraenkel axioms.

Thus they are now rarely considered as axiomatic as such.

However, in their time they were groundbreaking.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 12$: The Peano Axioms - 1964: J. Hunter:
*Number Theory*... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers - 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 1$: Numbers: $1.1$ Natural Numbers and Integers - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Peano's axioms**