I am trying to understand what is the meaning of Axiom 2.1of a book.
Is "0 is a natural number" same as "the origin is 0 and 0 is unique"
The reason I am asking is as follow.
Given:
Axiom 2.1: 0 is a natural number
Axiom 2.2. If n is a natural number, then n++ is also a natural number.
Axiom 2.3. 0 is not the successor of any natural number; i.e., we have n++ = 0 for every natural number
Axiom 2.4. Different natural numbers must have different successors; i.e., if n, m are natural numbers and n = m, then n++ = m++. Equivalently2, if n++ = m++, then we must have n = m.
Trying to transform the "natural number object" from [natural number: name and successor] to [natural number: name and ancestor]. Is this transformation valid, however? Does every natural number beside 0 have an ancestor? Can we define multiple different set of natural number start from different origin such that not natural number beside 0 have an ancestor?