Only bleem is not the secret integer between 3 an 4 but the smallest real number greater than zero. Yes, I know real analysis and all about supremum and coverings and epsilons and all the other ways to hide its existence. John Conway gets close in "On Numbers and Games" (there is some of the secret in non-standard analysis) but the conspiracy pushed him off into knot theory when he got too close... Still if you drop the real number line from (0,1) something has to hit the ground first. Bleem.
No, no, no. See, I think it's between 0.99999999_ (repeating) and 1. I mean, something has to go between the two, or people will think they're secretly the same number. But they're not. Numbers can't be Superman. They can't just change costumes in the blink of an eye. They should always look the same, dammit.
Sure, you can multiply 1/3 (== 0.33333_) by 3 and make it look like they're the same, but there's really a tiny little epsilon of something that's getting zeroed out.
It's a conspiracy I tell you! They hide all the cool stuff with infinity and you can't see it any more, then it's like it never existed.
It took thousands of years to accept that the parallel postulate might not be required, and lots of neat geometry emerged. Why not have number systems where infinitesimals are allowed?
Actually, if you look at the original version of that page (http://en.wikipedia.org/w/index.php?title=Archimedean_proper...), it doesn't mention the word "axiom" at all. Someone edited the page at some point to introduce this concept! It seems to have been done in a somewhat anonymous fashion, lending credence to Natsu's theory that there's a conspiracy involved here.
Yep, exactly :). (Should have made it more clear that I was referring to other systems like surreals and hyperreals, but wasn't clear whether the parent was joking or not).
Thanks for the pointer! I enjoy discussions about .999... because it really makes us question what we mean about infinity (and assumptions about the reals).
In mathematics, a definition cannot be 'wrong'; it can lead to an inconsistency, but that isn't the same thing.
The definition of the real numbers (the set of numbers where 0.999... == 1) does not lead to any inconsistency. However, there are other consistent definitions of sets of numbers where (the equivalent of) 0.999... does not equal (the equivalent of) 1. Those sets of numbers have values mathematicians call 'infinitesimals', which do not exist in the set of the real numbers. The hyperreals are one such set of numbers.
