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  • Presburger and by Skolem (both in 1930) that arithmetic with addition alone or multiplication alone is decidable (with regard to truth) and therefore has complete formal systems.

  • 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).

  • 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.

  • We introduce a new mode of question, which enables one to ask about a Skolem function for an operator. xhere denotes a finite sequence of variables x1, …, xn, and ∀xstands for ∀x1…∀xn.

  • 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.

  • 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.

  • To make a logic that fits these games, we use the same first-order language as in the previous section, except that a notation is added to some quantifiers (and possibly also some connectives), to show that the Skolem functions for these quantifiers (or connectives) are independent of certain variables.

  • 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).

skolem sentence examples

  • 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.

  • It is thus remarkable that von Neumann's work, designed to show how the transfinite ordinals can be incorporated directly into a pure theory of sets, builds on and coalesces with both Kuratowski's work, designed to show the dispensability of the theory of transfinite ordinals, and also the axiomatic extension of Zermelo's theory suggested by Fraenkel and Skolem.

  • 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.

  • 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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    Use skolem in a sentence

    On this page, there are 20 sentence examples for skolem. They are all from high-quality sources and constantly processed by lengusa's machine learning routines.

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    • 17 sentence examples for skolem from Stanford Encyclopedia of Philosophy
    • 3 sentence examples for skolem from Encyclopedia Britannica