Many, in which the items with the same properties

(A) the number is characterized not only by quantity, but also the properties of numbers.
The axioms of Peano say nothing about the properties of numbers. I need a definition of the set in which all elements with the same properties, but there are no axioms on the properties of the numbers.
What do I need it? Certain actions, (part of) the items in the set changes the properties, but to reveal it is possible only by way of reasoning, there is no mathematical approach for this.