Kline, M. (1980). “Mathematics: The Loss of Certainty” Oxford University Press: Oxford.

Morris Kline on the Löwenheim-Skolem theorem, pg. 271

“Suppose one were to set down a list of the features which, he believes, characterize Americans and only Americans. But, surprisingly, someone discovers a species of animal which possesses all of the characteristics on the list but also totally different features from Americans. In other words, axiom systems that are designed to characterize a unique class of mathematical objects do not do so….the Löwenheim-Skolem theorem tells us that a set of axioms permits many more essentially different interpretations than the one intended. The axioms do not limit the interpretations or models. Hence mathematical reality cannot be unambiguously incorporated in axiomatic systems.”



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