On Fri, 01 Jun 2007 01:10:57 -0400, Bob Kolker <firstname.lastname@example.org> wrote:
>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.
But one needs to use different terms. You can't call your new numbers "the reals" and the standard real numbers also "the reals". I'm not objecting to defining new types of numbers, I'm objecting to the ambiguity of terminology. For example, you can have
"the standard reals", "the hyper-reals", etc.
but in any given discussion, at most one of them can be abbreviated as just "the reals". By default, "the reals" means "the standard reals", unless declared otherwise.