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Please dont jump on me before reading it all. I am aware and l agree that Within the Standard Reals 1=0.999.....

Now, I know only a bit about the Hyperreals and other non-standard models of the Reals. From the little I know, the equality 1=0.9999... holds within the Hyperreals if the sequence is indexed by the Hyper Integers but not if it is indexed by the Standard Natural numbers. Is this correct? It seems we could find a fixed hyperreal that would be larger than the difference 1-0.999..., but I don't know enough to "rigorize" the statement. Can anyone answer these questions and add anything else you believe is relevant or helpful?

MSIS
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    The comments on the following (very similar) question might help: https://math.stackexchange.com/questions/3653526/reference-request-how-is-0-99-cdots-defined-in-nonstandard-analysis – Z. A. K. May 22 '20 at 16:36

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Let's make our notation more explicit.

First, let's briefly recap the standard situation. Decimal representations are really just infinite sums, and in particular $$0.9999...:=\sum_{i\in\mathbb{N}}{9\over 10^i}$$ (I'm using the convention that $0\not\in\mathbb{N}$ here). There's an implicit claim here: that that infinite sum exists in the context we're working in (the standard real numbers). While "obvious" this is actually nontrivial - for example, even nicely-bounded and all-terms-positive infinite sums need not make sense in $\mathbb{Q}$ (consider $3.14159...$).

Now let's look at the nonstandard situation. Surprisingly, "naive" sums are harder now! The length-$\mathbb{N}$ sequence $$0.9,0.99,0.999,...$$ does not have a supremum in the$^1$ hyperreals, and so "$\sum_{i\in\mathbb{N}}{9\over 10^i}$" does not make sense in nonstandard analysis.

However, this is because we've mixed up notions: we're bringing the $\mathbb{N}$ from standard analysis into the universe of nonstandard analysis, and this doesn't work. The hyperreal universe has its own kinds of sequences and series, which are no longer indexed by $\mathbb{N}$ but rather by $^*\mathbb{N}$, the nonstandard natural numbers. Very very informally, this means that nonstandard analysis' version of "$0.9999....$" has "infinitely deep" digits. This notation makes sense within the hyperreal context ... and by exactly the usual argument, equals $1$. Of course I haven't tried to define what a sequence/series of nonstandard length "really is;" this is a topic that you need to dive into nonstandard analysis to see in detail, and I don't think I can do it justice here.

So basically the situation is the following: when we jump from standard to nonstandard analysis, we do see a distinction between $0.9999...$ as normally construed and $1$, but this is because our normal construal of $0.9999...$ is inappropriate for nonstandard analysis and doesn't actually name anything specific at all.

$^1$There isn't actually a single thing called "the hyperreals;" rather, there's a general notion of hyperreal field, and in nonstandard analysis we work in some hyperreal field. Excluding really esoteric topics, the specific choice of hyperreal field doesn't matter and so we often ignore it.

Noah Schweber
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  • Thanks. Are you referring to the Compactness/Lowenheim-Skolem when you say there is no such thing as "The Hyperreal field"? Can you give me some refs on how to deal with indexes for the sequence other than Hyperintegers? – MSIS Nov 18 '20 at 21:58
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    @MSIS I'm not sure exactly what you mean. My point is just that there are lots of hyperreal fields out there, so that it doesn't really make sense to talk about "the hyperreal field." – Noah Schweber Nov 18 '20 at 21:59
  • But are these hyperreal fields isomorphic/ elementary-equivalent ( aka transfer principle) to each other, of different cardinalities? – MSIS Nov 18 '20 at 22:02
  • Thanks. I may not be making too much sense, not being knowledgeable in this area. – MSIS Nov 18 '20 at 22:11
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    @MSIS They are not isomorphic, but they are all elementarily equivalent (since by the transfer principle each is elementarily equivalent to $\mathbb{R}$ itself). – Noah Schweber Nov 18 '20 at 22:12
  • Hi Noah. My (recent) understanding is that hyperreal numbers are defined as equivalence classes of sequences of real numbers. So we could regard 0.999... as the equivalence class of the sequence (0.9, 0.99, 0.999, ...) and this is indeed infinitesimally less than *1. Would you say that's a reasonable way to look at it? – Rivfader Feb 29 '24 at 08:06