- #51

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The ambiguity is in the "..." -- since the decimal number system was invented long before Cantorian ordinals, the "..." is the "naive infinity" which is usually assumed to by omega, the smallest transfinite ordinal. If you state that assumption explicitly, then there is nothing to discuss, since you have then defined the notation precisely enough that the you can apply Cauchy's theory of limits to it and prove that 0.999... = 1. End of story.

However, if you allow the decimal to continue to a higher ordinal number of decimal places, you have a notation that seems not to correspond to the standard reals. Of course, this does not by itself create a number system for the notation to point to. (If that were true, then Anshelm's ontological proof of the existence of God would actually be valid and we should all be convinced by it. I think not.)

There are of course various extensions of the reals, particularly the hyperreals and surreals, which seem (at first glance anyway) to be candidates for number fields that would be appropriately expressed by such a larger-ordinal notation. As far as I am aware, this is an open question (someone correct me if I am wrong). I do know that Conway and Berlekamp showed that one particular notation (the "Hackenbush game") for the surreals had a subset of expressions that had an obvious correspondence with the real numbers expressed in the binary system.

I'm not sure that this contribution to the thread advances the discussion in any way; I'm just trying to add something that is perhaps of interest to a discussion on a topic that is extremely tedious to mathematicians who have to deal with cranks on occasion.

--Stuart Anderson