Would the axioms be able to replace Peano’s axioms?

0 is a natural number.

There are no natural numbers before 0.

There is no last natural number, every natural number has a successor.

For every natural number n, n = n: every natural number is equal to itself.

For all natural numbers n and x, n = x and x = n: equality is symmetric.

For all natural numbers n, x, and y, if n = x and x = y, n = y: equality is transitive.

For every natural number n, if n = x, then x is a natural number: every natural number can only be equal to a natural number.

For every natural number n, S(n) is the successor of n.

If n is a natural number, then the S(n) is also a natural number.

For all natural numbers n and x, if S(n) is S(x) then n is x: all successors are unique.

For every natural number n, S(n) = 0 is false: there is no natural number whose successor is 0.

For every natural number n, n+x is S(n)...x: addition is recursively succeeding on a natural number the amount of times a natural number.

For every natural number n, n+x is y: any natural number can be reached by adding a natural number onto a natural number.

For every natural number n, n+0 is n: adding 0 to any natural number results in the natural number.

For every natural number n, n*x is n+x...x: multiplication is recursively adding a second natural number onto a natural number the amount of times the second natural number.

For every natural number n, n*x is y: any natural number can be reached by multiplying a natural number onto a natural number.