On Thursday, February 27, 2014 11:45:33 AM UTC-5, John Gabriel wrote: > On Thursday, 27 February 2014 17:46:32 UTC+2, Dan Christensen wrote: > > > On Thursday, February 27, 2014 10:28:33 AM UTC-5, John Gabriel wrote: > > > > > So, we do NOT have 2+2=4? > > > > Pay careful attention! At your website, you say this: > > > > "The Definition of the set of natural numbers is given by nothing more or less than Peano's Axioms." >
By this, I meant that the Peano axioms give us the essential properties of the natural numbers from which, it would seem, all other known properties can be derived (apart from perhaps Godel's variations of the Liar Paradox). In that sense, they define the natural numbers however you may want to visualize them.
> > > That is outright bullshit. The Peano axioms assume the prior existence of natural numbers.
Not really. By themselves, the Peano axioms are just statements about a hypothetical function on a hypothetical set. Like the group or ring axioms in this regard.