But if you agree that whatever .999… is, it has the same value both in the reals and the hyperreals, then why do you care about the hyperreals? The problem can be attacked in the reals and the result transferred to the hyperreals or vice versa.
Which is exactly why the hyperreals are a tremendous distraction here.
If all results were identical for reals and hyperreals, then there would be no point in having hyperreals. And he has already referred to differences in convergence/divergence.
Interesting discussions using non-standard approaches to the question are here :
cut-the-knot.org/WhatIs/Infinity/9999.shtml
and here :
Not at all (you wish).
If neither you nor Ed3 really can’t come up with a proof (and wtf certainly can’t), I’ll go ahead and show you the truth of the matter. You won’t accept it of course, because you never do, but the reasoning will be valid. That is why I wanted you to think about it first.
I didn’t bring it up.
I don’t think I need to work through the countability of the rationals.
Okay, I brought it up. But it’s the only way that you’re going to stick an infinitesimal difference into the equation - it’s the only way for the equations to balance. ( You can’t say that 1/3 =/= 0.333… without breaking fundamental operations in the Real system.)
The reals have no bottom number from which to jump to zero. Everyone is talking as if they do. And if you think they do, please tell us what it is.
Nor do the hyperreals. If ε > 0 is an infinitesimal in the hyperreals then so is ε/2 and 0 < ε/2 < ε. This is true because the hyperreals are a field (just as the reals are) and you can always divide any element by 2.
There is no smallest positive hyperreal. This is another illustration of the transfer principle. There is no smallest positive real therefore there is no smallest positive hyperreal. They satisfy the same first-order statements. They are both models of the exact same axioms.
You don’t need to do anything you don’t want to do. But if you work through the Wiki page on the transfer principle then we can talk about why .999… = 1 is a theorem either in both the reals and the hyperreals, or neither of them.
The hyperreals are a technical construction in model theory, a branch of mathematical logic. If you’re not willing to do some serious math, there’s no hope of presenting rigorous proofs in a forum post. It’s advanced math, far more comple than the mere countabiilty of the reals. But if you accept the transfer principle, the .999… = 1 is either a theorem of the reals and the hyperreals, or neither.
Since it’s easy to show that it’s a theorem of the reals (I have done so many times) it’s a theorem of the hyperreals.
You have a tendency to pick of various ideas from different systems and stick them together. Sometimes that’s interesting and innovative and sometimes it produces a mangled mess.
In the Real number system, 1/3=0.333… and you can’t get away from that without going to non-standard analysis with another number system.
I’m not going to waste more time going round and round about it.
Nobody is writing a PhD thesis here so we don’t need “rigorous proofs”. Take a hint from one of my links : "A friendly chat about whether 0.999… =1 ". Same advice for James. 
You just proved why 1 =/= 0.999…, in both systems. 
And you just proved that you are unwilling or unable to engage with a substantive argument either in math OR philosophy.
In any event there is already a proof that .999… = 1 in the reals. It’s been presented many times in this thread and many times online. It’s a standard proof. And because it uses only first-order concepts (quantifying over individuals and not subsets or predicates) it’s equally valid in the hyperreals.
So there are two takeaways:
.999… = 1 in the reals and the hyperreals.
But even if you don’t believe that, waving your hands about the hyperreals doesn’t change the argument or the conclusion. It’s just a way of confusing yourself.
I linked you to the Wiki page on the transfer principle. It’s a technical result. Its proof is far beyond the scope of this discussion.
But we don’t have to understand its proof. We can simply accept its conclusion. Any first-order statement true in the reals is true in the hyperreals. A first-order statement is one where the quantifiers only talk about individual numbers and not sets of numbers. .999… = 1 is a first-order statement. So it’s provable in both the reals and the hyperreals or neither. That’s a friendly proof sketch. It’s as good as we can do without discussing the technical details.
The friendly point being that the hyperreals don’t add anything to the discussion. If someone can prove that .999… = 1 is invalid in the hyperreals then they can pull back the proof to get the same result in the reals.
So the hyperreals are irrelevant. That’s the relaxed, friendly, non-technical explanation.
And yet any number of folks on this thread – the objectivists – would seem to suggest that in fact it is either one or another.
And that makes sense to most folks because in the realm of mathematics it always seems to work out that way. For example, Nat Geo has a series on now about Albert Einstein – Genius. Mathematics abounds in it. But the whole point seems to be that his calculations are either in sync with is in fact true or they are not. That he is closer to whatever reality actually is than, say, Newton.
Yet wouldn’t it be extraordinary if it turned out that it might be one or the other. That, with respect to the reality of, among other things, dark matter, dark energy, the quantum world etc., mathematical computations are very different?
Again, for folks like me this is not clearly understood. We are more curious instead about how the world that we live in from day to day would be impacted – for all practical purposes – if the mathematics was interpreted [used] in another way.
How, for example, would the engineers who invent the new technologies that we just take for granted be impacted by the shift?
They have to. After all, philosophers are concerned not only with whether 1 = 0.999 or not, but how the answer to that question is integrated into an understanding of the very nature of reality and existence itself. Just as theologians are concerned with how the answer to that questions is integrated into an understanding of God.
For me, however, it is always the same bottom line: What in the course of human interaction can be known wholly, fully…and what cannot?
My main focus is on the world of “is/ought”. But what if it is no less problematic of the either/or world in turn?
That, perhaps, there is no either/or world?
Is it all really that “spooky”?
You’re confusing math with physics. Even JSS has agreed that we are talking about math here and not physics. There is no physical referent for the real numbers. In physics we use the real numbers to approximate things. There is no claim in contemporary physics that the mathematical real numbers have any correspondence to anything in reality. Nor could there be, since in physics we can not sensibly discuss anything below the Planck scale, while in the real numbers we can divide a number in half as many times as we want till its value goes far below the Planck scale.
.999… = 1 is a question of math, not physics. You could not design a physical experiment to test the proposition since no physical measurement can ever be sufficiently precise.
Perhaps.
But in watching the series Genius, I suspect that most folks would be hard put to note with any precision where mathematics ends and physics begins in Einstein’s probing of the universe. And the laws that encompass it.
This sort of argument may well be “technically” correct. It is certainly over my head. But once mindless matter managed to evolve into mindful matter – matter able to probe itself – it would seem all that more problematic [“spooky”] to explain the importance of answering questions like this.
Why should it matter one way or another?
Does it matter one way or the other?
And now at last we are discussing the philosophy of mathematics! Yes you raise vital questions. I don’t know the answer. If I say that math is only a formal game played with meaningless symbols according to arbitrary rules, then why is math so “unreasonably effective” in the natural sciences, as Wigner put it.
On the other hand if I say that math = physics, then an expression like .999… makes no sense at all since we can’t measure below the Planck scale.
I don’t know the answer but it’s a heck of a good question. Perhaps start a separate thread. It has no relation to the question of .999…, which can ONLY be answered mathematically. It has no referent in the real world. We know that.
Based on which definitions you start with, one interpretation may be entirely correct or may be more or much more reasonable.
Sets of equations model reality and so some sets may be more accurate over an entire range of conditions or over a limited range of conditions. If you need a particular accuracy, then you use a particular mathematical model. You use Newton’s equations for motion 99% of the time because they are good enough and simpler than the Einstein stuff.
An engineer decides how much accuracy is required in the process of solving a problem. He/she gets training on how many digits in a number are valid and significant. Nobody works with decimal numbers which have infinite or a huge number of digits. These numbers are truncated or rounded off at some stage. Prior to that they are manipulated as symbols or fractions - which is why we have a symbol specifically for Pi.
But there is that aspect of dasein and subjective judgement - maybe in this particular context more digits would be better - personal gut instinct of the engineer. 
One can know stuff well enough … or not. Again, judgement call. Ultimately you test your calculated result in the real world.
You and I have different ideas about what ‘friendly’ means. Nobody has a life at stake here, or millions of dollars or anything of significance. So…
Maybe I want to think about this problem in a particular way. Maybe I want to go down the wrong path and eventually end up on the right path. Maybe it’s not the wrong path. Maybe I want to get lost.
That means that I don’t necessarily want someone to hold my hand and to offer me guidance, instructions and the one correct way of thinking.
Do you get what I’m saying, professor? Daddy?
I would be perfectly happy to answer, to the best of my ability, any specific questions you have. I’m no expert on the hyperreals and learned a lot of what I know after JSS brought them up last year and I spent some time diving into technical materials about them. They are very murky, much more so than people realize. Please ask any questions you have.
I might not be understanding your point. I’m not a professor and I’m not your daddy. You are not required to read my posts, nor respond to them. You can ignore me as you like, or ask me questions and I’ll be happy to answer the best I can.
But now that you mention it, many people associate someone being mathematically precise, with some awful authoritarian grade school math teacher who rapped your knuckles with a ruler when you couldn’t remember your times tables. That’s why (I think) JSS talks about the “holy prophets” in reference to people who simply write math textbooks.
I state for the record that I can’t take any responsibility for anyone’s early experiences with awful math teachers, even though I do deplore the state of math education. You’ll have to get over it if you’re going to participate in a discussion about math.