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). A unit is a ratio of equal magnitudes.
So, by axiom 3, 2 is that number whose difference with 1 is 1.
Again, by axiom 3, 3 is that number whose difference with 2 is 1.
Any other numbers you want to derive from my axioms? :-)