(Skolem 1922: appendix.)
Results such as those obtained by Gödel and Skolem were unmistakably semantic—or, as most logicians would prefer to say, model-theoretic.
It is a kind of Downward Löwenheim-Skolem Theorem for second-order logic.
Tarski refers his readers to a paper of Thoralf Skolem in 1919 for the technicalities.
On the other hand, the upward Löwenheim-Skolem theorem in its usual form fails for all infinitary languages.
No Completeness, Compactness, or Löwenheim-Skolem theorem is available for second-order logic with standard models.
As early as 1922 Skolem speculated that the CH was independent of the axioms for set theory given by Zermelo in 1908.
The other is by taking unions of elementary chains, generalising the proof we gave for the upward Löwenheim-Skolem theorem.
Among these Tarski himself would mention Gödel’s completeness theorem and several versions of the Löwenheim-Skolem theorem (cf.
Second, it can be regarded as the first place where recursive definability is linked to effective computability (see also Skolem 1946).
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.
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).
The depressing news is that there are no categorical first-order theories with infinite models; we can see this at once from the upward Löwenheim-Skolem theorem.
Lindstrom has shown that the Löwenheim-Skolem theorems play a key role in characterizing first-order logic itself (Lindström 1966; Lindström 1969; Ebbinghaus 2007).
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.
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.
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.
Once this preliminary argument is complete, the Skolemite can then proceed to use the algebraic conception of sets (plus, of course, the Löwenheim-Skolem theorems) to defend the claims about set-theoretic relativity that are made in steps 2 and 3 of his argument.
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.
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).
Skolem
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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 Zermelos 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