On 2/27/2014 3:03 PM, Virgil wrote: > In article <firstname.lastname@example.org>, > email@example.com wrote: > >> My book gives the correct axioms. >> >>> For over a century now, I think Peano's axioms have been the standard, >> >> Like Cantor's nonsense. Where do they define that the difference must always >> be 1? > > They don't. They only requires that each member of a Peano set have a > unique successor member, but do not require any specific 'difference' up > front. > It is only later, after all the basic axioms are satisfied, when > defining addition that any "1" appears. >
That is actually problematic.
Willard's work in arithmetic looks to systems that focus on division and difference rather than mulitplications and additions.
But, Mr. Gabriel has it wrong. In the axioms that he did not understand in multiple languages, there is a subtle definition of even numbers. Even numbers are those measured by 2 equal parts. Odd numbers are those that are not.
Then, Euclid observes that an odd number differs from an even number by a unit.
Equals are compared with respect to even numbers in the sense that equals measure even numbers by reflection as a part.
So, there is a natural grouping of pairs of odd numbers with their enclosed even number congruent to 2 mod 4 into triples. The "difference" between consecutive odd numbers is 2 equal parts which are units.
Now, if the triples are interleaved with the numbers congruent to 0 mod 4, then the "difference" between extrema for consecutive triples is 2 equal parts which are not units.
Notice how I am using "equals" "units" and "not units". The number labels make it easier to understand the model, But, once you understand the model, the labels can be obfuscated.