# Per (Pelle) Lindström

This is a much condensed version of the obituary by Väänänen and Westerståhl in Theoria 2010 (76) pages 100-107.

Per Lindström, or Pelle Lindström as he insisted on being called, was born on April 9, 1936, and spent most of his academic life at the Department of Philosophy, University of Gothenburg, in Sweden, where he was employed first as a lecturer (‘docent’) and, from 1991 until his retirement in 2001, as a Professor of Logic.

Lindström is most famous for his work in model theory. In 1964 he made his first major contribution, the so-called Lindström’s test for model completeness. In 1966 he proved the undefinability of well-order in L_{ω1ω} (obtained independently and in more generality by Lopez-Escobar). The same year he also introduced the concept of a Lindström quantifier, which has now become standard in model theory, theoretical computer science, and formal semantics.

It was his 1969 paper ‘On extensions of elementary logic’ (in Theoria), where he presented his famous characterizations of first-order logic—Lindström’s Theorem—in terms of properties such as compactness, completeness, and Löwenheim-Skolem properties, that was first recognized as a major contribution to logic. It laid the foundation of what has become known as abstract model theory. The proof was based on Ehrenfeucht-Fraïssé games, a concept he came up with independently, and on a new proof of interpolation. Several other characterizations of first-order logic followed in later papers.

Beginning at the end of the 1970’s, Lindström turned his attention to the study of formal arithmetic and interpretability. He started a truly systematic investigation of this topic, which had been somewhat dormant since Feferman’s pioneering contributions in the late 1950’s. In doing so he invented novel technically advanced tools, for example, the so-called Lindström fixed point construction, a far-reaching application of Gödel’s diagonalization lemma to define arithmetical formulas with specific properties. Pelle Lindström had an exceptionally clear and concise style in writing mathematical logic. His 1997 book, Aspects of Incompleteness, remains a perfect example: it provides a systematic introduction to his work in arithmetic and interpretability. The book is short but rich in material.

Throughout his life, Pelle Lindström also took an active interest in philosophy. He participated in the debate following Roger Penrose’s new version of the argument that Gödel’s Incompleteness Theorems show that the human mind is not mechanical. He presented his own philosophy of mathematics, which he called ‘quasi-realism’, in a paper in The Monist in 2000. It is based on the idea that the ‘visualizable’ parts of mathematics are beyond doubt (and that classical logic holds for them). He counted as visualizable not only the ω-sequence of natural numbers but also arbitrary sets of numbers, the latter visualizable as branches in the infinite binary tree, whereas nothing similar can be said for sets of sets of numbers, for example.

Pelle Lindström passed away in Gothenburg, Sweden, on August 21, 2009, after a short period of illness.