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The theorem that bears their names—the Löwenheim-Skolem theorem—has two parts.
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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.
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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.
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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.
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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.
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Model theory can be regarded as the product of Hilbert's methodology of metamathematics and the algebra of logic tradition, represented specifically by the results due to Löwenheim and Skolem.
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On the other hand, the upward Löwenheim-Skolem theorem in its usual form fails for all infinitary languages.
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No Completeness, Compactness, or Löwenheim-Skolem theorem is available for second-order logic with standard models.
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In the light of the geometric example, Weyl attacks the concept “defined by means of finitely many words” as not precise, and long before Fraenkel and Skolem, he succeeds in making the separation principle precise: he simply replaces Zermelo’s informal concept of definite property with the notion of ‘relation explicitly definable from extensional equality and membership by means of the basic elementary logical principles’ (we should simply say: first-order definable).
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(Skolem 1922: appendix.)
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Turning to the Löwenheim-Skolem theorem, we find that the downward version has adequate generalizations to L(ω1,ω) (and, indeed, to all infinitary languages).
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Skolem in the winter of 1915–16 visited Göttingen, where he discussed set theory with Felix Bernstein; there is no sign that he met Hilbert.
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Tarski refers his readers to a paper of Thoralf Skolem in 1919 for the technicalities.
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Second, it can be regarded as the first place where recursive definability is linked to effective computability (see also Skolem 1946).
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The metatheory of second-order logic with Henkin models is very much like the metatheory of classical quantificational logic in that it is complete, compact, and subject to a Löwenheim-Skolem theorem.
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Second, we should note that, although this approach requires the Skolemite to start with an independent argument against our ordinary conception of sets, it need not render the Löwenheim-Skolem theorems themselves completely superfluous.
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Besides Russell himself, and despite all these complications, Chwistek tried to develop arithmetic in a ramified way, and the interest of such an analysis was stressed by Skolem.
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Button (2011) has argued that, although this kind of technical criticism has teeth against the version of Putnam's argument which explicitly invokes the downward Löwenheim-Skolem theorem, there are alternate formulations of Putnam's argument which can evade the criticism.
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Lindstrom has shown that the Löwenheim-Skolem theorems play a key role in characterizing first-order logic itself (Lindström 1966; Lindström 1969; Ebbinghaus 2007).
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As early as 1922 Skolem speculated that the CH was independent of the axioms for set theory given by Zermelo in 1908.
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