It is, for you, who is sufficiently sophisticated to find grounds on which to have an actual debate on the matter. But it’s not for the posters coming in here with some ‘proof’ that 0.999~ differs from 1. When the biologically naive say ‘Darwinian evolution isn’t true’, it’s something entirely different from when the evolutionary biologist says ‘Darwinian evolution isn’t true’—he will presumably have in mind all the little ways in which we have amended the theory to arrive at the modern synthesis, and thus, be right, while the creationist is just wrong.
A number of things, some of them not quite wholesome even if you leave out the pure trolls.
Some people need someone to validate their pet theory, which belongs to them, and is theirs. Back in the Handwritten Age, they sent long, disjointed, rambling letters to university departments of mathematics, all of which seemed to be in green ink. (The letters, not the departments.)
Others are motivated by the same thing that motivates me: I want to quash ignorance where I see it, especially in places like this. It niggles at me. The problem, of course, is that in this instance they’re simply wrong. Not true, in fact, which is a much stronger statement than merely being wrong, which leads me into…
The third group are incapable of understanding that, in mathematics, it’s possible for some things to be utterly and completely wrong with no possible hint of a possibility of ever being even partially correct or, as I said above, not true. Saying that, in the field of the reals (defined as everyone else defines it), 0.999… ≠ 1 is not true, and that just irks them beyond belief, apparently. It irks them at a deep, philosophical level that mathematics can be absolute. I even remember a poster in a previous thread claiming that this isn’t a mathematical issue, but a philosophical one, and that (apparently) the position which is not true would be saved by a technical subtlety that mere mathematicians were unable to grasp. Modified versions of the position can be so saved, but the position as stated cannot be.
A sub-group of the third group are ultrafinitists turned the wrong way: They have a deep philosophical objection to the idea of any idea extending forever and ever, or even much beyond their bedtime. Zeilberger they’re not, though, because they fail to separate notation from concepts, so they say that “0.999… comes close to 1, but it isn’t 1” and similar nonsense, because they fail to grasp that notations mean what we’ve defined them to mean, and that 0.999… has a perfectly good definition which gives it a perfectly finite value.
I hope the above does not come across as an attack, it’s certainly not meant as one. You know (I hope) how much I value your contributions in this thread and the boards in general. But maybe sometimes it’s better to shelve discussions on flying until everyone’s clear on the mechanics of walking :-). Not that I always heed that advice…
I would append onto that their incomprehension of, anger toward, and total bewilderment about the concept of infinity.
They don’t know or care about set theory, they refuse the complications and rigor of Cantor. They want common sense. IOW, they want arithmetic to handle infinity. And they are very, very firm about this.
No problem. I myself worry about coming across as attacking, and when my pedantry is useful vs. pointless, and this and that and the other thing all the time. In fact, I was about to write a similar “I hope that didn’t come across as an attack…” post to you!
(Also, in this particular thread, I do worry whether I’m doing more enabling than enlightening when saying “Of course, you could interpret the notation as…” (and “Let’s see if we can find a way to formalize your intuitions” and so on) instead of “Listen, bub, .999… = 1; get it through your head!”; to ignore the former altogether would be to disingenuously shelve the truth, but have I hit the right ratio between them in terms of emphasis? I don’t know. I suppose it depends on the target audience, and in a thread this long, it’s no longer clear who that is.)
Well, it’s not really any single ‘audience’, but more a continuous stream of challengers, which forces the discussion to undergo periodic resets, at least partially.
I understand your concern about unfairly ‘shelving’ the truth—I’ve had a similar situation in some recent Einstein-sucks-ether-rules thread: there is certainly a possibility to meaningfully introduce a concept of ether into modern science, and there are modern approaches like Einstein-ether theory or condensed matter analogies, fluid spacetimes and the like (which I happen to find very interesting, even though it’s not really my field), but what your typical ether-revivalist means can’t be usefully connected to what modern-day ether theory (such as it is) is about, because their own misconceptions stand in the way of a fruitful generalization of their notions. At the time, I decided to try and address these misconceptions at their base, rather than opening the door to further misunderstandings; not that I could claim that my efforts were terribly fruitful, so hey, what do I know.
(EDIT: Oh dear, now I’m responsible for this thing to have grown another page…)
What is an axiomatization of a number line? Does it just determine which numbers are bigger than which numbers?
I should probably leave that for somebody who actually knows what they’re talking about, but basically, the axiomatization tells you the stuff about ordering you mentioned, how to do multiplication and addition, and then something that gives you the completness of the reals—completeness here not being meant in the logical sense that every valid formula is a theorem, but rather in the sense that there are no ‘holes’ between real numbers, which can be formalized in various ways, via Dedekind cuts, or by requiring that every (Cauchy-) sequence converges, i.e. that any sequence of numbers such that the distance between any two successive numbers shrinks beneath any given value has a unique limit they approach, or various other ways. (I think that’s basically it…)
What I’m trying to figure out is how you could describe addition and multiplication in the reals without ipso facto having done so for the naturals*–and I thought that being able to do this with the naturals was sufficient for goedel stuff to apply.
*i.e. the model of the reals would have the naturals in it as a “sub model” so to speak, wouldn’t it? This is probably where I’m going wrong as I’m not defining “sub model” explicitly either here or in my own head.
Yeah, once you’ve defined addition on the reals, it’s just a notational definition to use “S(n)” as a substitute expression for “n+1”… so everything in arithmetic on the naturals can be translated into a statement in the mathematics of the reals—so the latter is “powerful enough to express the arithmetic of natural numbers” isn’t it? ISN’T IT?
For years I keep thinking I understand these things only to see someone make an offhand comment that pulls me up short.
If you limit your language enough, you can’t define the naturals as a subset of the reals, and thus can’t translate statements quantifying over naturals into statements only quantifying over the reals.
For example, if we, as mentioned above, limit our language to first-order statements about polynomial equations, then we will find that every definable set of reals is simply a finite union of intervals. (The jargon name for this property is “o-minimality”). The naturals are clearly not of this form, and thus we will not generally be able to translate the relevant sorts of natural arithmetic into this language.
As noted above, if we add, for example, cosine to our language, then we will be able to define the naturals as a subset of the reals, and will be able to carry out such translation, bringing along all the corresponding Goedelian phenomena.
I would have said, ‘you can’t define a predicate that picks out the integers among the reals’, but the above is probably OK, too. 
One important thing to say is that “Why is .999… = 1?” is not a stupid question. (It may have been said before in this thread, but I’m not going to check that by reading or re-reading 1,000+ replies.)
It’s not stupid, because the answer is not simple. It involves how mathematicians define the real numbers, limits and infinity. It involves ancient paradoxes, and how they have been resolved. And it’s the subject of a Wikipedia article that’s about 10,000 words long (0.999…, already cited above.)
Are we going to have a birthday party for this thread … in some cultures, 14 year-olds are considered adults. That would Wednesday after this.
I’d like to say, though, that it’s not really that systems of natural (first-order + and *) arithmetic are subject to the Goedelian phenomena and systems of real (first-order + and *) arithmetic escape it. Both are subject to the same limitation on their ability to consistently prove certain statements about themselves; it’s just that the failure occurs even earlier in the case of real arithmetic, in that the relevant statements are not even expressible in the language we might choose to limit our attention to in the latter case.
There’s also nothing particularly special about the language of natural arithmetic, except that it happens (by a tremendous hack) to be capable of discussing the behavior of arbitrary computable functions, and we happen to be particularly interested in computable formal systems; accordingly, the systems we are particularly interested in are describable in terms of natural arithmetic, so if the language against we judge their completeness itself includes natural arithmetic, it is capable of expressing the relevant Goedelian statement which the system cannot hope to prove.
But any framework which speaks about itself in the appropriate way, whether it has anything to do with natural numbers or not, and whether it is computable or not, displays all the same Goedelian phenomena in all the same ways as natural arithmetic does; the former has no intrinsic connection to the latter.
And you also can’t pick out just every real at a fixed interval? (n, n+m, n+2m, n+3m, etc?)
There’s actually an interesting observation lurking here: It’s trivial that the integers are naturals are interdefinable over any context where ordering is definable (the naturals being the integers >= 0; the integers being the values which are either naturals or negated naturals).
I’ve refrained, for parsimony, from mentioning ordering explicitly in discussing the relevant language of real arithmetic so far, but even if not adopted as a primitive of our language (though in fairness it should be, for the clean development of the theory of “quantifier elimination” behind the completeness results), we can, in the reals, define ordering from the observation that a real is >= 0 just in case it has a real square root.
But suppose, instead, we were interested in the language of first-order integer + and * (without a primitive for ordering). Can we define the naturals within these (and therefore ruin any chance of algorithmically deciding truthhood and falsehood within this language)?
Yes, we can: an integer is a natural just in case it is the sum of the squares of four integers. But, alas, this is much harder to prove than our characterization of positivity within real arithmetic.
If your language is suitably expressive, sure.
But if your language is just =, >, 0, +, 1, *, Boolean operations, and first-order quantifiers over the reals, how would you propose doing so?
Right, what I meant was I am surprised you can’t do it with just those tools. But realizing you can’t explains to me how it could be that you’ve modeled the reals yet not ipso facto also “overmodeled” the naturals.
What’s the actual axiomatization of the reals that we’re talking about? What are the axioms? I guess I don’t really understand this unless I’ve seen the axioms I get to work with to try to prove theorems that map onto the peano axioms.