The Peano axioms were meant to provide a rigorous foundation for the natural numbers (0, 1, 2, 3,...) used in arithmetic, number theory, and set theory.
Only thing is that the natural numbers already had a rigorous foundation - from Euclid's Elements. In fact, the rational numbers were rigorously in place from Euclid's Elements. The 5 step derivation I provided of rational numbers is how Euclid did it. There really was no need for monkey Peano to formulate his juvenile "axioms". You don't provide rigour by using all the properties and construction of the object whose foundation you are attempting to solidify.
The Natural numbers as well as the Rational numbers have a very rigorous foundation in Euclid's Elements. Peano was an idiot (like most of you here) who did not understand the difference between magnitude or number.
A magnitude is NOT a number. Euclid defined magnitudes in Book 5 and numbers in Book 7. Book 7 contains the construction of both natural and rational numbers.
Go back now and read my 5-step construction of rational numbers.