On Wednesday, 26 February 2014 19:45:27 UTC+1, Dan Christensen wrote: > 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.
The chracteristic property of the natural numbers is their constant distance of 1 from their next neigbours. That feature distinguishes them from all other Peano-sequences.
> 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. > But not as the defnition of the natural numbers. The natural numbers can only be defined by the natural process of counting or of adding units, i.e., by the foundation of mathematics. That's why we call them natural numbers. It means to turn bottom on top if not the natural process of counting is used as the base of all mathematics but some unprecise axioms should serve to derive the most natural mathematical act. Perversion pure. Some people seem to need that.