On Thursday, February 27, 2014 11:28:57 PM UTC-5, John Gabriel wrote: > On Friday, 28 February 2014 06:17:30 UTC+2, Dan Christensen wrote: > > > > > So, this is another axiom? > > > > Of course yes. It's by construction of the natural numbers. >
OK, that's 9 "axioms of arithmetic" now.
> > > > How about the associativity of + and *? Two more axioms? How about their commutativity? Yet two more axioms? How about their cancelability, etc., etc..... > > > > Those properties are proved after the construction of natural numbers. >
I'm confused. Haven't you already constructed the natural numbers with you 8, oops... make that 9 "axioms of arithmetic"? And how do to propose to prove this without induction?
> > > Peano's rot does not even construct the natural numbers.
There you go again, John. Again, Peano's axioms describe the essential properties of the natural numbers. In formal mathematics, these properties can be used to define the natural numbers.
> It assumes they exist with ALL their properties. The successor function is an assumption of the property that two successive natural numbers have, that is, a unit difference. > > > > Do you honestly think that Peano's "axioms" are anything more than a juvenile attempt on his part? :-) >
I think they are a thing of beauty. Embrace them, John. ;^)
> > > > I think you will need some equivalent of the Peano's 5th and most important axiom: the principle of induction. You really can't do number theory without it. > > > > You don't [need?] Peano's rot.
I think you will find that you do need them, John.