Definitions and nondefinability in geometry. (English) Zbl 1205.51010

Smith, James T.

Am. Math. Mon. 117, No. 6, 475-489 (2010).

A model of good expository writing, this is a historical presentation of the axiomatics of plane Euclidean geometry, with particular emphasis on the primitive notions the language in which the axiom system is expressed contains. In lucid and very readable prose, the author first presents the axiomatic method as understood by the Greeks, followed by its modern rebirth in the works of Pasch, Peano, Pieri, and Hilbert. Here the author dwells upon the reason why Hilbert’s Grundlagen der Geometrie of 1899 overshadowed all other work on the subject, in particular why it condemned M. Pieri’s [Mem. di Mat. e di Fis. Soc. It. Sc. (3) 15, 345–450 (1908; JFM 39.0545.08)] to the fate of not even being considered worthy of a Jahrbuch review.
Besides the significant difference in the position Hilbert and Pieri occupied in mathematics as perceived by their contemporaries, attributable to Hilbert’s many achievements in various areas of mathematics, the author discerns another reason for the neglect of Pieri by the general mathematical readership: the fact that, unlike Hilbert, Pieri did not use defined notions in his presentations, and provided complete, detailed proofs for all statements, making his papers appear to be very long logical treatises. However, precisely the aspect that was considered cumbersome by the general, not logically-minded reader, made Pieri’s work interesting to one of the 20th century’s greatest logicians, Alfred Tarski, who provided — influenced by Pieri’s axiom system in terms of a single ternary relation \(I\), with \(I(abc)\) to be read as ‘\(ab\) is congruent to \(ac\)’ — his own remarkably simple axiom system, based on the notions of betweenness and equidistance.
Here the author mentions that Pieri’s axiom system is the simplest possible one from two points of view: (i) it is formulated in terms of a single ternary relation among individual variables to be interpreted as ‘points’, and no system of binary relations can serve as primitive notions for Euclidean geometry [as shown by A. Lindenbaum and A. Tarski [Ergebn. Math. Kolloq. Wien 7, 15–22 (1935; JFM 62.0039.02)], (ii) it can be rephrased so that each of its axioms are \(\forall\exists\)-statements, as shown in the reviewer’s [Rend. Semin. Mat., Univ. Politec. Torino 67, No. 3, 327–339 (2009; Zbl 1223.51028)].
After proving by means of Padoa’s method two undefinability results, namely that equidistance cannot be defined in terms of betweenness, and the Lindenbaum-Tarski theorem referred to above, the author points out that, as shown in the reviewer’s [Zesz. Nauk., Geom. 18, 5–8 (1990; Zbl 0722.51014); Zesz. Nauk., Geom. 19, 87 (1991; Zbl 0729.51011)], the binary predicate of unit distance alone can serve as primitive notion for plane Euclidean geometry over Archimedean ordered Euclidean fields if the axiom system is expressed in the infinitary logic \(L_{\omega_1\omega}\), which allows for denumerably infinite conjunctions and disjunctions.

Reviewer: Victor V. Pambuccian (Phoenix)

MSC:

51M05	Euclidean geometries (general) and generalizations
03-03	History of mathematical logic and foundations
51-03	History of geometry
01A60	History of mathematics in the 20th century
01A55	History of mathematics in the 19th century
03C40	Interpolation, preservation, definability
03B30	Foundations of classical theories (including reverse mathematics)

Keywords:

Pieri; Tarski; definability; axiom systems for geometry; undefinability; plane Euclidean geometry; equidistance; betweenness

Citations:

Zbl 0722.51014; Zbl 0729.51011; Zbl 1223.51028; JFM 39.0545.08; JFM 62.0039.02