On Sunday, March 2, 2014 11:42:04 AM UTC-5, John Gabriel wrote: > On Sunday, 2 March 2014 17:07:49 UTC+2, Dan Christensen wrote: > > > > > > > You are being evasive, John Gabriel. But since you ask, you don't formally "arrive at" any axioms. You just state them, and move on to deriving other statements from them -- something which, it seems, you are incapable of doing with your system of "axioms". > > > > That's what I thought. You were asking me to do the impossible. :-) Now that you've realised this, let me add that you don't just make statements and call them "axioms". Axioms are a special class of statements that have been shown to be true. :-) > > > > So, my axioms are true statements. :-) > > > > > I derived the entire RATIONAL NUMBERS. > > > You just said those were axioms! You don't formally "derive" axioms. > > > > I said that I derived the rational numbers using my axioms. I did not say that I derived the axioms. :-) Very different things, but it would help if you could read properly. > > > > > > > Again, it seems you cannot derive even the most elementary results of number theory, e.g. the existence of 2 distinct numbers (see my proof posting here) > > > > That is a lie. Any rational numbers can be derived from my axioms. > > > > > It seems you have painted yourself into a corner, John Gabriel. > > > > On the contrary, it is you who are in the corner. :-) > > > > > > 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. > > > > 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 magnitude that measures either number in terms of the other. > > > > 5. If a unit is divided by a number into *equal* 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. > > > > Let's derive the numbers 2 and 3 using my axioms. > > > > We have already established the unit (1) from the axioms of magnitude (5 easy steps I showed you).
How did you "establish" that? Which of your "axioms" did you use? Please explain how you used those axioms to "establish" this result.