# Model Theory: Robinson

# MODEL THEORY: ROBINSON

Abraham Robinson (1918–1974) was a logician and mathematician. Born in Waldenburg (Silesia), he moved to Palestine in 1933, where he studied mathematics at the Hebrew University in Jerusalem and also joined the Haganah. In 1940 he fled to Britain as a wartime refugee and enlisted with the Free French Air Force. He took his PhD in London in 1949 while teaching aerodynamics at the Cranfield College of Aeronautics. He held posts successively in Toronto, Jerusalem, Los Angeles, and finally Yale, where he died of cancer. His eventful life is described by Joseph W. Dauben (1995).

Robinson's PhD thesis on applications of logic in mathematics led to an invitation to speak at the International Congress of Mathematicians in 1950. The talks of Robinson and Alfred Tarski at this congress became founding documents of the new discipline that Tarski named model theory. Throughout his career Robinson was one of the most fertile contributors of programs, techniques, and results to model theory.

Robinson's thesis contains his independent discovery of the compactness theorem for first-order languages of any cardinality. In the proof he introduced constant symbols to stand for the elements of the model to be constructed. He noticed that if these constant symbols corresponded to the elements of a given structure *A*, and the theory contained sentences expressing all the relations of the structure *A*, then any model of the theory would contain an isomorphic copy of *A*. This observation became the method of diagrams, which Robinson used systematically as a way of creating models of a theory with prescribed embeddings between them. Diagrams immediately became one of the fundamental techniques of model theory (for many applications, see Robinson 1963).

Robinson switched from one branch of mathematics to another with extraordinary ease. There were certain topics that he kept returning to from different angles. Two in particular were elementary embeddings and algebraically closed fields. Combining the two, he noted that every embedding between algebraically closed fields is elementary. He coined the term *model-complete* for theories whose models have this property and devised tests to show when a theory is model-complete.

Observing the role of algebraically closed fields in field theory, he looked for analogous structures within other classes. Model completions, model companions, infinite forcing companions, and finite forcing companions were notions that he proposed at various times as generalizations of algebraic closure. He identified the classes of real-closed fields and differentially closed fields as the model completions of the ordered fields and the differential fields, respectively, and axiomatized the class of differentially closed fields (though the usual axioms are an improved version due to Lenore Blum). In 1965 the notion of model completion played a central role in the proofs by James Ax and Simon Kochen, and independently by Yuri Ershov, of a number-theoretic conjecture of Emil Artin.

Around 1960 he noticed that any proper elementary extension of the field of real numbers contains infinitesimals. He quickly developed this insight into a powerful and intuitively natural approach to mathematical analysis that he named nonstandard analysis. Nonstandard analysis is one of the few innovations in logic that were entirely the work of a single individual.

Not long before his death, Robinson collaborated with the number theorist Peter Roquette to apply model-theoretic methods in number theory. This work gave a first hint of the deep interactions between model theory and diophantine geometry that came to light in the 1990s, sadly too late for Robinson to contribute. In fact, Robinson died before he could take on board the stability theory pioneered by Michael Morley and Saharon Shelah, though his students, Greg Cherlin and Carol Wood, did contribute to this field, bringing with them Robinson's lifelong eagerness to apply model theory to algebra, algebraic geometry, and mathematics in general.

Though unable himself to believe in any kind of existence for infinite totalities, he strongly defended the right of mathematicians to proceed as if such totalities exist. His discussion (Robinson 1965) of mathematical and epistemological considerations that favor one or another of the traditional views in philosophy of mathematics is thoughtful but seems not to reveal a thoroughly worked out position. His anti-Platonistic attitude may have helped him to create nonstandard analysis by allowing him to be relaxed about what the "real" real numbers are.

In Robinson's *Selected Papers* (1979), the bibliography lists ten books, more than a hundred papers, and a film. One in seven of his papers are in wing theory and aeronautics.

** See also ** Infinitesimals; Model Theory; Tarski, Alfred.

## Bibliography

Dauben, Joseph W. *Abraham Robinson: The Creation of Nonstandard Analysis—A Personal and Mathematical Odyssey*. Princeton, NJ: Princeton University Press, 1995.

Robinson, Abraham. "Formalism 64." In *Proceedings of the International Congress for Logic, Methodology, and Philosophy of Science, Jerusalem 1964*, edited by Y. Bar-Hillel, 228–246. Amsterdam, Netherlands: North-Holland, 1965.

Robinson, Abraham. *Introduction to Model Theory and to the Metamathematics of Algebra*. Amsterdam, Netherlands: North-Holland, 1963.

Robinson, Abraham. *Selected Papers*. 3 vols, edited by H. J. Keisler et al. Amsterdam, Netherlands: North-Holland, 1979.

*Wilfrid Hodges (2005)*

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