On 2/26/2014 3:17 PM, email@example.com 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. >
I had been referring to the remark that you might manage to get the *actual* natural numbers from among the rest with a little bit of good luck.