 
Summary: The Birth of Model Theory: L¨owenheim's Theorem in the Frame of the
Theory of Relatives
by Calixto Badesa
Princeton, Princeton University Press, 2004
xiii + 240 pages. US $52.50. ISBN 0691058539
Reviewed by Jeremy Avigad
From ancient times to the beginning of the nineteenth century, mathemat
ics was commonly viewed as the general science of quantity, with two main
branches: geometry, which deals with continuous quantities, and arithmetic,
which deals with quantities that are discrete. Mathematical logic does not
fit neatly into this taxonomy. In 1847, George Boole [1] offered an alter
native characterization of the subject in order to make room for this new
discipline: mathematics should be understood to include the use of any
symbolic calculus "whose laws of combination are known and general, and
whose results admit of a consistent interpretation." Depending on the laws
chosen, symbols can just as well be used to represent propositions instead
of quantities; in that way, one can consider calculations with propositions
on par with more familiar arithmetic calculations.
Despite Boole's efforts, logic has always held an uncertain place in the
mathematical community. This can partially be attributed to its youth;
