A constructive version of Tarski's geometry

Abstract

Constructivity, in this context, refers to a theory of geometry whose axioms and language are closely related to ruler and compass constructions. It may also refer to the use of intuitionistic (or constructive) logic, but the reader who is interested in ruler and compass geometry but not in constructive logic will still find this work of interest. Euclid's reasoning is essentially constructive (in both senses). Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive (in both senses), even if we restrict attention (as we do here) to the theory with line–circle continuity in place of first-order Dedekind completeness. Hilbert's axiomatization has a much more elaborate language and many more axioms, but it contains no essential non-constructivities. Here we exhibit three constructive versions of Tarski's theory. One, like Tarski's theory, has existential axioms and no function symbols. We then consider a version in which function symbols are used instead of existential quantifiers. This theory is quantifier-free and proves the continuous dependence on parameters of the terms giving the intersections of lines and circles, and of circles and circles. The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Pasch's axiom and the parallel axiom; this choice of function symbols connects directly to ruler and compass constructions. All three versions have this in common: the axioms have been modified so that the points they assert to exist are unique and depend continuously on parameters. This modification of Tarski's axioms, with classical logic, has the same theorems as Tarski's theory, but we obtain results connecting it with ruler and compass constructions as well. In particular, we show that constructions involving the intersection points of two circles are justified, even though only line–circle continuity is included as an axiom. We obtain metamathematical results based on the Gödel double-negation interpretation, which permit the wholesale importation of proofs of negative theorems from classical to constructive geometry, and of proofs of existential theorems where the object asserted to exist is constructed by a single construction (as opposed to several constructions applying in different cases). In particular, this enables us to import the proofs of correctness of the geometric definitions of addition and multiplication, once these can be given by a uniform construction.We also show, using cut-elimination, that objects proved to exist can be constructed by ruler and compass. (This was proved in [3] for a version of constructive geometry based on Hilbert's axioms.) Since these theories are interpretable in the theory of Euclidean fields, the independence results about different versions of the parallel postulate given in [5] apply to them; and since addition and multiplication can be defined geometrically, their models are exactly the planes over (constructive) Euclidean fields.