On Monday, March 3, 2014 12:47:06 AM UTC-5, iwa...@gmail.com wrote: > On Sunday, March 2, 2014 7:44:02 PM UTC+2, Dan Christensen wrote: > > > > > Huh??? If asked you how many noses you have, and you said "I have foot nose!" (or "metre nose") how much sense would that make? You are grasping at straws, John Gabriel. > > > > Well, I established the unit and the natural numbers in my axioms. >
You did not such thing. Neither your "unit" nor your 0 are explicitly assumed to be a number. Call it "obvious" if you will, but it is NOT in your "axioms".
> > > What did you do in yours? You established nothing, not even ZERO. >
I have established the existence of 2 distinct numbers using only the first 3 Peano axioms. Of course, 0 is assumed to be a natural number in this case. I show that must exist another non-zero natural number.
The Peano axioms are well know to provide a foundation for all of number theory and analysis. And your "axioms" as they stand, John Gabriel? You can't even prove the existence of 2 distinct numbers.
Your "axioms" really need a bit of tweaking. You will have to actually define the essential characteristics of numbers, the meaning of "larger" and "smaller", etc. if you are to salvage them. You really aren't gaining anything with that silly commutative difference relation. Better, I think, to start with either addition or some kind of successor function. But then you might as well use the time-tested Peano axioms!
By all means continue grasping at straws, John Gabriel. This is great fun!