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The study of non-standard models did not start with Gödel’s results—Skolem, in particular, was already aware of them earlier in a different context (he had discovered that first-order theories of set theory have unnaturally small, namely, countable models, in Skolem 1922; cf. the entry on Skolem’s paradox)—but the first incompleteness theorem elucidates the existence of non-standard models in the context of arithmetic, while the nonstandard models elucidate the first incompleteness theorem.
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As for set theory, the failure of categoricity was already taken note of by Skolem in 1923, because it follows from the Löwenheim-Skolem Theorem (which Skolem arrived at that year; see Skolem 1923, based on Löwenheim 1915 and Skolem 1920): any first order theory in a countable language that has a model has a countable model.
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These technical results were of great importance for the subsequent debate over first-order logic.
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We have Soundness and Completeness theorems for classical quantificational logic with identity as well as Compactness and Löwenheim-Skolem theorems.
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It is not only the Löwenheim-Skolem theorem but also other metamathematical theorems can be given a paraconsistent treatment.
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Among these Tarski himself would mention Gödel’s completeness theorem and several versions of the Löwenheim-Skolem theorem (cf.
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It is a kind of Downward Löwenheim-Skolem Theorem for second-order logic.
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The functions that spell out the dependencies of variables on each other in a sentence of first-order logic were first considered by Skolem and are known as Skolem functions.
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Results such as those obtained by Gödel and Skolem were unmistakably semantic—or, as most logicians would prefer to say, model-theoretic.
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Since the principles of arithmetic, analysis and set theory had better possess at least one infinite model, the Löwenheim-Skolem theorem appears to apply to them.
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Or so, at any rate, Jané takes Skolem to be arguing.
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In the 1922 paper where he originally presented Skolem's Paradox, Skolem used the paradox to argue for two philosophical conclusions: that set theory can't serve as a “foundation for mathematics” and that axiomatizing set theory leads to a “relativity of set theoretic notions” (Skolem 1922).
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Thus, the rudimentary treatment of mathematics in the Tractatus, whose principal influences were Russell and Frege, was succeeded by detailed work on mathematics in the middle period (1929–1933), which was strongly influenced by the 1920s work of Brouwer, Weyl, Hilbert, and Skolem.
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If every structure had a saturated elementary extension, many of the results of model theory would be much easier to prove.
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Thus this proof of the Completeness Theorem gives also the Löweheim-Skolem Theorem (see below).
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But at the same time, many of the basic definitions and results in recursive function theory are only indirectly related to recursive definability in the informal sense described in this section.
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Taking these principles as a foundation, Skolem showed how to obtain recursive definitions of the predecessor and subtraction functions, the less than, divisibility, and primality relations, greatest common divisors, least common multiples, and bounded sums and products which are similar to those given in Section 2.1.2 below.
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The name of the theorem is a little unfortunate, since the theorem was first proved by Tarski, and Skolem didn’t even believe it (because he didn’t believe in uncountable cardinals).
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The Compactness Theorem was extended to the case of uncountable vocabularies by Maltsev in 1936 (see Mal'cev 1971), from which the Upward Löwenheim-Skolem theorem immediately follows.
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But Skolem never mentions the fact that the existence of such models follows from the completeness and compactness theorems.
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