Calixto Badesa. The Birth of Model Theory: L wenheim's Theorem in the Frame of the Theory of Relatives Princeton: Princeton University Press, 2004. Pp. xiii + 240. ISBN 0-691-05853-9.
2005; Oxford University Press; Volume: 13; Issue: 1 Linguagem: Inglês
10.1093/philmat/nki004
ISSN1744-6406
Autores Tópico(s)History and Theory of Mathematics
ResumoWhen we encounter a theorem with a composite name, like Heine-Borel, Cantor-Bendixson, or Löwenheim-Skolem, we are curious to know what the particular contribution to it of each author actually was.The obvious guess is an alternative: either the first author provided a deficient or incomplete proof, or else the second author generalized the original theorem.As regards the Löwenheim-Skolem theorem, both things are the case.The theorem was first proved in 1915 by Leopold Löwenheim (1878Löwenheim ( -1957)), and then reproved and generalized by Thoralf Skolem (1887-1963) in 1920, in 1922, and again in 1929.As stated by Löwenheim in 1915 and by Skolem in 1920, the theorem says that if a first-order sentence has a model, then it has a countable (finite or infinite) model.On the deficiencies of Löwenheim's proof something will be said later, but for now it is worth noting that in 1920 Skolem did not claim that Löwenheim's proof was defective-only that some aspects of it were 'somewhat involved' (van Heijenoort [1967a], p. 254) and he wanted to give a simpler proof.As to the generalization that Skolem provided, it consisted in extending the theorem so as to apply to a countably infinite set of sentences instead of a single one.Calixto Badesa's monograph The Birth of Model Theory deals, as its subtitle says, with Löwenheim's theorem in the frame of the theory of relatives.It concentrates on the first two sections of Löwenheim's 1915 essay Über Möglichkeiten im Relativkalkül, 1 where first-order logic is singled out for study and Löwenheim's Theorem is proved.Löwenheim's arguments are carried out in the theory of relatives, initiated by Augustus De Morgan and extensively developed by Charles S. Peirce and Ernst Schröder.The theory of relatives is discussed in the second chapter of the book, the first being devoted to a concise account of the development of the algebra of logic from George Boole to Löwenheim's time.The rest of the book (chapters 3 to 6, and an appendix) focusses on Löwenheim's proof.
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