On Wednesday, February 26, 2014 12:43:12 PM UTC-5, muec...@rz.fh-augsburg.de wrote:
> The Peano axioms fail to produce this set because they are too general. They produce the set 1, S1, SS1, ... > > Some of the many sets produced are > > 1, 1s, 1ss, ... > > 1, 11, 111, ... > > 1, 1^1, 1^1^1, ... > > and if someone happens to have defined, or simply to know, the natural numbers, then they produce also > > 1, 1/2, 1/3, ... > > 1, 1^2, 1^3, ... > > and with good luck > > 1, 2, 3, ... >
Assuming the implicit successor functions in each case, these sets are all isomorphic to one another, i.e. you are simply using different symbols for the various elements, but the successor relation is preserved, as is the induction principle. As I said, such structures is embedded in EVERY infinite set. So, you are on fairly safe ground to assume the existence of one such structure at the beginning of a proof.