TITLE: Term for "uncheckable constructions"
QUESTION [30 upvotes]: Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge construction which checks that this is a right answer. (We assume that we can see if two points coincide.)
On the other hand Squaring the Circle is not checkable --- there is no way to say "yes" by performing a finite Compass-and-straightedge construction.
Similarly one can not check that a given point is a center of given circle by straightedge only. (Here I assume that straightedge can not draw infinite lines, so one can not check whether two lines are parallel.)

REPLY [5 votes]: Quick claims about Joel's definition:
Constructibly decidable, semi-decidable, and co-semi-decidable are identical for constructions which, like most classic geometric constructions, take bounded numbers of steps. (By the same proof that those decidable, semi-decidable, and co-semi-decidable are identical with a bound on the computation time.)
A constructible semi-decidable problem is exactly a countable disjunction of bounded constructible decision problems, and similarly for co-semi-decidable and conjunction. (Assuming there is no computably criterion in the definition of a geometric construction that takes unbounded time.)
We can classify the constructibly decidable in bounded time problems using the Euclidean geometry - algebra correspondence. Representing each point by a pair of numbers, any geometric input can be represented as a tuple of numbers. The constructible functions are exactly the compositions of rational functions and square roots.
To get a yes-or-no value rather than a number as the result of our computations, we need a test. We can test whether two points are equal, so we can test equality, and we can test whether a line is on a circle, so we can test inequality.
Any finite decision procedure is thus described by a logical combination of formulas where you take a function generated by rational functions and square roots and test whether it is positive. Such a decision procedure is clearly a first-order statement in the theory of real closed fields (if we take a real closure of our number line). By quantifier elimination, every first-order statement in the theory of real closed fields is given by a decision procedure, and you don't need to mess with square roots.
So a constructibly decidable problem is a partition of $\mathbb R^n$ into two sets, each a countable union of first order definable sets.
Need such a partition be a partition into two first order definable sets? No. For instance the set of real numbers whose integer part is even is a countable union of the intervals $[2n,2n+1)$, which are first order definable, but it is not definable, and similarly for its complement.