On Friday, 28 February 2014 07:17:41 UTC+2, Martin Shobe wrote:
> >> I agree that 6 - 4 = 2 wouldn't show that 2 + 3 = 6.
> Yes 2 + 3 = 6 is false. That's why the fact that it fits your axioms is > such a problem.
You do not understand the first axiom.
The difference (or subtraction) of two numbers (2 and 3) is that number (6) which describes how much the larger exceeds the smaller.
So why can't you have 2 + 3 = 6?
Because, 6 - 4 = 2, but we KNOW that the numbers are 2 and 3, not 2 and 4.
Can you have 6 - 3 = 3?
No. Because you KNOW that the numbers are 2 and 3, not 3 and 3.
Moreover, you cannot say 1 - 3 or 0 - 3 because the smaller is subtracted ('-') from the larger.
The first axiom establishes the primitive operator '-' from which all the remaining operations are derived.
> Your axiom was, "The difference (or subtraction) of two numbers is that > number which describes how much the larger exceeds the smaller." Three > is most definitely two larger than one, so 1-3 = 2. Similarly, three is > three larger than 0, so 0 - 3 = 3.
You don't do this even in the flawed Peano axioms. What makes you think you can do it in mine? The axiom states clearly the smaller must be subtracted from the larger: "Difference" is defined as "how much the larger exceeds the smaller"