Dan Christensen <Dan_Christensen@sympatico.ca> wrote in news:email@example.com:
> 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!
What he appears to be doing (and failing) is constructing numbers from measurements of magnitudes.
It is something that could possibly be done, but it is so sloppy and ambiguous that its nowhere it yet. He confuses magnituted, measurements, and numbers. He assumes numbers exist in order to perform measurements based on some multiples of an arbitrary unit and then claims that that means he is constructing numbers. He also defines rationals (the only numbers he things exist) in terms of a fraction, but this also presupposes numbers exit. He fails to define most of the key terms. He defines a unit as both any magnitude that you can compare to itself and as a ratio of a magnitude to itself and confuses these two concepts subsequently. And we both know that his definition of arithmetic operations are very poorly worded, giving valid interpretation where 2 + 3 = 6 etc. Those definition could be so very easily fixed, but he just can't even see that there is a problem.
You're right though, its great fun watching him trying to justify the unjustifiable without ever dealing with the valid criticisms, whereas if he put a little work into it, he could at least have the start of a resonable set of axioms and definition.