  We provide a variant of an axiomatization of elementary geometry based on
logical axioms in the spirit of Huzita--Justin axioms for the Origami
constructions. We isolate the fragments corresponding to natural classes of
Origami constructions such as Pythagorean, Euclidean, and full Origami
constructions. The sets of Origami constructible points for each of the classes
of constructions provides the minimal model of the corresponding set of logical
axioms.
  Our axiomatizations are based on Wu's axioms for orthogonal geometry and some
modifications of Huzita--Justin axioms. We work out bi-interpretations between
these logical theories and theories of fields as described in J.A. Makowsky
(2018). Using a theorem of M. Ziegler (1982) which implies that the first order
theory of Vieta fields is undecidable, we conclude that the first order theory
of our axiomatization of Origami is also undecidable.
