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No, I have not altered the order type. All I have done is use a bijection between $\mathbb{N}$ and the ordered set. In $\omega+1$, $\omega$ is still the largest element -- but it's the one you insert first. So in your example we could start by mapping $\omega$ to $0$, then map $0$ to $-1$, then $1$ to $-1/2$ etc. The insight is that one doesn't have to insert your elements in increasing order!
It does work: $\omega+1$ is a countable set. We can write its elements as $\omega,0,1,2,\ldots$. This is an order we can insert them in. There is no "final" entry.
At first sight, this looked nice, but now I'm not sure. This $f$ is constant in the unit disc, but while $f$ is not an analytic extension of $f$ from the unit disc to $\mathbb{C}$ the constant function $F(z)=1$ is.
If Qiaochu's interpretaion is right, each of these equidistance conditions defines a plane, and generically three planes meet in a unique point. I'm not sure how this squares with Alok's final sentence about the "best" point and spheres which seem to be looking for something like the point whose greatest distance from the given points is minimized.
Again, I look in vain in one of Pavlov's links for any clues about the analytic work that hides behind his cohomological conisderations. His previous comment explicates neither in what sense polynomials are dense within smooth functions (nor any proof that they be really are) nor any other inegration-free proof of Poincare's lemma. But in that thread another correspondent revealed that Poincare duality can be proved via Hodge theory, so there may be an even more expensive way of avoiding some simple integration theory than Pavlov advocates.
Pavlov asserts that the surjectivity of $d$ can be proved by the density of polynomials in smooth functions (yet he declines to specify in which topology) and that more details are in "the link". The only link specified in this thread is to lecture notes of Schapira. Schapira provides no details of this density argument, and his argument for the surjectivity of the operator $\partial_{j+1}$ (required in the proof of the Poincare lemma) consists of the single word "Clearly" (last line of page 123).
I can't figure out what you mean: $Ax+By$ whatever it is, is not a "binary operation". So does "$A=2$, $B=1$" mean that if $x$ and $y$ are elements of your set, then $2x+y$ is?
