Edit: Magdiragdag's response clarifies something along the lines of what I was thinking
The canonical proof I'm referring to is the one shown in Artin Algebra (15.5) or the one found in these notes: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gao.pdf
Here is the part where I'm encountering trouble:
These rules seem really restrictive to me, because as far as I can tell, they don't allow for the construction of arbitrary points satisfying certain properties. (see edit: clarification in bold letters)
It is often the case that euclidean construction problems often involve constructing arbitrary points, lines, and circles. (I don't have an example currently at hand but many Euclidea problems can only be done by constructing an arbitrary object)
I thought that the construction of arbitrary points would eventually be addressed in the ongoing proof, but it never happened. All constructions strictly adhered to the rules shown above. And when the proof ended, I felt thoroughly betrayed because the proof completely ignored the case where you can construct arbitrary points as you wish. Isn't this what is naturally done in practice?
My question: is there a way to formalize the concept of "constructing an arbitrary point satisfying some parameters" and integrate it into the canonical proof?
Edit (clarification): the rule set above provides a deterministic existence-consequence "chain reaction" which is how we end up constructing the field extensions, but how can we address the case where we simply add an arbitrary point into the picture? Distinguished from constructed points, arbitrary points needn't always be constructed points.
I want to show that with the added freedom of being allowed to construct arbitrary points, lines, and circles, that trisecting an angle is still deterministically impossible (i.e. there is no general method that will guarantee successful trisection each time the method is repeated).
I have some ideas of what this entails, but I can't combine them into one coherent proof. Here are some facts that can be deduced from the rules of ruler and compass construction above, given (0,0) and (1,0) are constructed points:
given the coordinates of some arbitrary point A, one can construct a point B that is arbitrarily close to A using the rules above
Moral fact: Let S be the set of the points on the x-y space satisfying a set of properties P (e.g. position relative to a line, etc.). If we can deterministically construct an arbitrary point such that we be can certain that it belongs to S, then S must be infinite. (e.g. Let S be the set of points above the constructed line y=x. Then we can construct an arbitrary point such that we are sure it belongs to S).
Another moral fact: The only properties that S can take are boolean: (e.g. above or below the line L, inside or outside the circle C, lies on or lies off line T)
I feel like a proof sketch would look something like this:
We wish to construct an arbitrary point in S satisfying the list of properties P.
We somehow prove that there exists a point in S with an open neighborhood. Thus, S must contain a constructed point from fact 1.
Suppose a method to trisect an angle exists and it involves constructing arbitrary points. Whenever we get to such a step where arbitrary point construction is required, there is always the chance we end up constructing a constructed point. In the case where we end up constructing a constructed point at every step involving arbitrary point construction, the proof becomes equivalent to the original canonical proof provided in Artin and the pdf.
Thus trisecting an angle while allowing arbitrary point construction is still deterministically impossible as was the case in the canonical proof.