quasi wrote: > > You claimed to have proved .999 is not equal to 1, so you've proved a > false statement. Without looking further, that already means that > means your proof is invalid. > > But I did look a little further. A quick scan of your proof shows you > are using your own alternative definitions for standard concepts. > Sorry, that's also cause for immediate rejection of your proof. You > are not allowed to do that -- read the rules.
One is allowed provided the basic undefined terms and the new postulates are clearly stated. This gives -another- system other than the standard real numbers. It in no way invalidates the standard real numbers. See any treatise on the hyperreals or non-standard analysis. > > I'm not saying that you are not allowed to attack the standards. You > can do that. Moreover, if you can show the standard concepts are > logically inconsistent, then those standards would have to be > abandoned. But to show such an inconsistency, you can't redefine the > concepts you are trying to attack. You have to use the existing > definitions, as they are already defined, and somehow prove a > contradiction. That's not what you did. You used alternative > definitions for standard concepts, and you reused the exact same > names. That renders your proof invalid.
Indeed. The theory of hyperreal numbers and non-archidean fields is yet another number system and another set of rules. All perfectly kosher provided it is consistent, which it is. When you remove the infinitesimals from the hyperreals you get back the standard real number system which is why the hyperreals constitute a kosher system and an interesting system at that. It shows why the calculus of Newton and Leibniz worked in the first place.