In article <email@example.com>, firstname.lastname@example.org wrote:
> On Wednesday, 26 February 2014 20:44:49 UTC+1, fom wrote: > > The set of natural numbers however is not every countable set but a very > > special countable set. 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, ... > > > > > > > A little bit of humor today? > > Humor? 1/3 - 1/2 is not 1. The set of natural numbers is not the same as all > sets isomorphic to Peano's set. You cannot define the natural numbers without > addition of 1. That's all.
One can define a set of natural numbers via its successor operation prior to any definition of "addition of one", and THEN define addition of one from that previously defined set of natural numbers and its previously defined successor operation.