On Friday, February 28, 2014 12:01:58 AM UTC-5, John Gabriel wrote: > On Friday, 28 February 2014 06:55:51 UTC+2, Dan Christensen wrote: > > > > > 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? > > Are you getting confused again between the construction of natural numbers and the axioms of arithmetic? I think you are! Chuckle. >
So, you have no idea how you are going to derive the basic rules algebra (associativity of +, etc.) in your system. You really have to resolve this issue, John.
> > > > There you go again, John. Again, Peano's axioms describe the essential properties of the natural numbers. > > > > Good. Now you are beginning to see the difference: the natural numbers are NOT the same as their properties.
In formal mathematics, their essential properties define them. You can go ahead and imagine the numbers arrayed on marble pedestals in heaven next to God, or some other such nonsense, but where would that get you? You need something you can work with.