John Gabriel <email@example.com> wrote in news:firstname.lastname@example.org:
> John Gabriel's construction of rational numbers from nothing in 5 easy > steps: > > 1. A magnitude is the idea of size of extent. We can either tell that > two magnitudes are equal or not. If we can tell they are not equal, > then we know which is smaller or bigger, but we can't tell how much > bigger or smaller. This is called qualitative measurement (without > numbers). > > 2. We can form ratios of magnitudes. AB : CD where AB and CD are line > segments. The expression AB : CD means the comparison of magnitudes AB > and CD. > > 3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to > use either as the standard of measurement, that is, the unit. The unit > is a ratio of equal magnitudes. > > 4. The unit enables us now to compare AB and CD if both are exact > multiples of the unit that measures both. We can now perform > quantitative measurement, because we can tell how much greater AB is > than CD or how much less AB is than CD. > > 5. Finally, if a magnitude is only part of a unit, then we arrive at a > ratio of numbers, say AB : CD where AB and CD are multiples of the > unit. AB : CD now means the comparison of numbers AB and CD. When we > write AB/CD, it is called a fraction. > > So, in five steps I have derived the concept of number for you. There > is one thing left - what happens when you can't measure a magnitude > that is not a multiple of a unit and can't be expressed exactly using > any part of a unit? This is called an incommensurable magnitude and > the best you can do is provide an approximation such as 3.14159... or > 1.414..., etc. > > Now that we have defined number, we state the axioms of arithmetic: > > John Gabriel's axioms of arithmetic: > > 1. The difference (or subtraction) of two numbers is that number which > describes how much the larger exceeds the smaller. > > 2. The difference of equal numbers is zero (same as no difference when > compared). > > 3. The sum (or addition) of two numbers is that number whose > difference with either of the two numbers is either of the two > numbers. > > 4. The quotient (or division) of two numbers is that number that > measures either number in terms of the other. > > 5. If a unit is divided by a number into parts, then each of these > parts of a unit, is called the reciprocal of that number. > > 6. Division by zero is undefined. > > 7. The product (or multiplication) of two numbers is the quotient of > either number with the reciprocal of the other. > > 8. The difference of any number and zero is the number. > > Observe that all the basic arithmetic operations are defined in terms > of difference, which is the most primitive operator.
Your definition of a unit is self-contradictory
And by your axioms you get 2 - 3 = 1, 2 + 3 = 6, 2 / 6 = 3, 0 - 3 = 3
You need to work on your axiom and number construction as they are rather sloppy and ambiguous.