[bubblyworld]

There's an extremely subtle point here about the hyperreals that the author glosses over (and is perhaps unaware of):

If you take 0.999... to mean sum of 9/10^n where n ranges over every standard natural, then the author is correct that it equals 1-eps for some infinitesmal eps in the hyperreals.

This does not violate the transfer principle because there are nonstandard naturals in the hyperreals. If you take the above sum over all naturals, then 0.999... = 1 in the hyperreals too.

(this is how the transfer principle works - you map sums over N to sums over N* which includes the nonstandards as well)

The kicker is that as far as I know there cannot be any first-order predicate that distinguishes the two, so the author is on very confused ground mathematically imo.

(not to mention that defining the hyperreals in the first place requires extremely non-constructive objects like non-principal ultrafilters)

[im3w1l]

So something I was thinking of: A number in decimal notation can be seen as a function from the integers to {0,1,2,3,4,5,6,7,8,9} (where negative numbers map to digits left of the decimal point and non-negative to digits right of the decimal point) such that only finitely many negative numbers map to non-zero.

Could you generalize this to include the hyperreals by lifting the restrictions on finitely many, and also adding in some transfinite ordinals to the domain of the function?

[bubblyworld]

I suspect yes - no need to introduce transfinite ordinals, you simply map from the set Z*, which is the integers but including the nonstandard ones. In fact you don't even need to remove the finiteness hypothesis, the transfer principle should guarantee that every hyperreal has such a representation since you can prove that every real does for the standard version.

(if the finiteness thing seems confusing, remember that there are infinitely large nonstandard integers in the hyperreals, and you can't tell them apart from the others "from the inside")