The Birth of Model Theory: Löwenheim’s Theorem in the Frame of the Theory of Relatives by Calixto Badesa (PDF)

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Löwenheim’s theorem reflects a critical point in the history of mathematical logic, for it marks the birth of model theory–that is, the part of logic that concerns the relationship between formal theories and their models. However, while the original proofs of other, comparably significant theorems are well understood, this is not the case with Löwenheim’s theorem. For example, the very result that scholars attribute to Löwenheim today is not the one that Skolem–a logician raised in the algebraic tradition, like Löwenheim–appears to have attributed to him. In The Birth of Model Theory, Calixto Badesa provides both the first sustained, book-length analysis of Löwenheim’s proof and a detailed description of the theoretical framework–and, in particular, of the algebraic tradition–that made the theorem possible. Badesa’s three main conclusions amount to a completely new interpretation of the proof, one that sharply contradicts the core of modern scholarship on the topic. First, Löwenheim did not use an infinitary language to prove his theorem; second, the functional interpretation of Löwenheim’s normal form is anachronistic, and inappropriate for reconstructing the proof; and third, Löwenheim did not aim to prove the theorem’s weakest version but the stronger version Skolem attributed to him. This book will be of considerable interest to historians of logic, logicians, philosophers of logic, and philosophers of mathematics.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a wonderful book, in which Badesa endeavours to make sense of the proof of Loewenheim’s theorem. In fact, what are Loewenheim’s fleeing indices? Badesa argues that we can give reason of them using the concept of family of set. It’s very interesting, because for the first time the fleeing indices are not seen as a sort of skolemian function. Wonderful is also Badesa’s effort to rebuild with a modern eye this important demonstration. Only one thing is to underpin: Badesa’s approach is very skolemian. It’s not clear to me, for example, if Loewnheim found a submodel D’ of the enumerable model D in which for hypothesis a formula of the first order is assumed valid. That is as Skolem saw Loeweheim….At any rate, it is a very important work in the history of the algebra of logic.

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