| Faelco said:
And yet that Timothy Gowers explains well that refusing that "convention" would mean "invent strange new objects" or "abandon some of the familiar rules of arithmetic". So he clearly accepts it. |
Hm? I don't understand your meaning. He accepts it as a convention, certainly. The idea that it's accepted because not accepting it would mean the system needs an overhaul does not invalidate the first part in the least. People are mostly bound by this system at this point and many don't think of it as a system but rather as reality itself. That's wrong.
Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[48] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[49] Lightstone shows how to associate to each number a sequence of digits,
indexed by the hypernatural numbers. While he does not directly discuss 0.999…, he shows the real number 1⁄3 is represented by 0.333…;…333… which is a consequence of the transfer principle. As a consequence the number 0.999…;…999… = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333…;…000…" and "0.999…;…000…" do not correspond to any number.
The standard definition of the number 0.999… is the limit of the sequence 0.9, 0.99, 0.999, … A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, …)] of this sequence in theultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number uH=0.999…;…999000…, with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality uH < 1.Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be
[50]
All such interpretations of "0.999…" are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….[51] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999… < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[52][53] Jose Benardete in his book Infinity: An essay in metaphysics argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:
The intelligibility of the continuum has been found—many times over—to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999… does not equal 1 but falls infinitesimally short of it. I think that .9999… should indeed be admitted as a number … though not as a real number.[54]There are other examples. I don't think I need to paste them all.
