While I fully agree with the correct subset of the arguments given above (you know who you are), it seems to me that there is a genuine ambiguity in the notation 0.999.... Caveat: Of course the real number system is defined axiomatically, so questions about the notation resolve ultimately not into questions about about the real number system per se, but about whether the notation is actually referring to the real number system or to some alternate system. This seems to be the difficulty with the original reference that started this thread.

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

Hurkyl
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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).
It's very easy for hyperreals -- hyperdecimal numbers are indexed by hyperintegers, just as ordinary decimal numbers are indexed by ordinary integers.

The hypernatural numbers are not (externally) well-ordered, but I don't see why the index set should be expected to be an ordinal.

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I wonder how this topic is really number theory...

It's very easy for hyperreals -- hyperdecimal numbers are indexed by hyperintegers, just as ordinary decimal numbers are indexed by ordinary integers.
Thanks! I had forgotten this (or never new, perhaps), as I have been more interested in the surreals than the hyperreals lately.

The hypernatural numbers are not (externally) well-ordered, but I don't see why the index set should be expected to be an ordinal.
No special reason why, I just mentioned larger ordinals as an example of a possible extension. BTW, could you give me a reference on the hypernaturals indexing the hyperreals? My acquaintance with this topic comes from Nelson's Radically Elementary Probability Theory, which went over non-standard analysis rather briefly as background for the application to probability.

Thanks!

--Stuart Anderson

Hurkyl
Staff Emeritus
Gold Member
BTW, could you give me a reference on the hypernaturals indexing the hyperreals?
It's a direct application of the transfer principle.

In the standard model, the decimal expansion of a real number s is nothing more than a function f:Z->{0, ..., 9} satisfying

$$s = \sum_{n \in \mathbb{Z}} f(n) 10^n$$
$$\lim_{n \rightarrow +\infty} f(n) = 0$$

and we have a theorem that every real number has a decimal expansion.

Applying the transfer principle tells us that in the nonstandard model, the hyperdecimal expansion of a hyperreal number s is nothing more than an (internal) function f:*Z->{0, ..., 9} satisfying

$$s = \sum_{n \in {}^\star \mathbb{Z}} f(n) 10^n$$
$$\lim_{n \rightarrow +\infty} f(n) = 0$$

and we have a theorem that every hyperreal number has a hyperdecimal expansion.

@Hurkyl

Got it! Thanks. I must be having a bad day, because it's obvious now that you mention it. Of course, the axioms are the same as for standard analysis when the new predicate "standard" is not used, so of course all the usual results follow whenever no distinction between standard and nonstandard values is made. This point was made clearly by Nelson, so I shouldn't have needed your prod. But apparently I did need it, so thanks!

--Stuart Anderson

0.9... is not =1. You can see
0.9.../=1(English)
My emai is
Changbai LI

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cristo
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