I have had this lingering doubt in my mind for a very long time: One of the standard constructions of N starts by assuming the 5 Peano Axioms, proving that every non-zero is a successor and s(n) is not equal to n for any n. Then one establishes the recursion theorem leading to addition, multiplication the usual laws, then order and trichotomy and finally WOP. Now most books "prove the equivalence of WOP and PMI and PCI."
My question is: While proving WOP implies PMI we need to assume that if n is not 0 then it is a successor. Are we justified?
My next question is what is the order one is considering. If it is the usual order then are we not assuming too much: Addition, Multiplication etc.?
It would be logically clearer if we knock PMI from PA, replace it with WOP with some total order plus one or two additional assumptions like 0 is the least and every non zero is a succesor and simply prove PMI. Is this possible?
Finally how would this new order relate to the usual order (since after PMI is established the usual game can be played.)
Answer will be very much appreciated.