Nice infinitary logics (Q2879888)

The paper concerns ``abstract model theory'', founded originally by Lindström's theorem which characterized first-order logics \({\mathcal L}_{\aleph_0, \aleph_0}\) as the nice logics satisfying the downward Löwenheim-Skolem-Tarski property to \(\aleph_0\) and the compactness theorem. Here, nice simply means that the logic considered has natural closure properties. The subject has somehow suffered for a long time from a lack of discovery of other logics which can be characterized in a reasonable way, and the author enthusiastically reopens the subject with new interesting examples. Lindström's theorem suggests that the nice theories not equivalent to first-order logic are the infinitary ones (usually above \({\mathcal L}_{\aleph_1, \aleph_0}\)) and the somewhat compact ones (usually \(\aleph_0\)-compact). The paper aims at presenting new infinitary logics with desirable properties, such as interpolation (known as Craig's theorem in the first-order case). As mentioned by the author, it does not deal with other major themes, such as \(\aleph_0\)-compact nice logics, logics without negation and continuous logic, or the question of isomorphism of models in some forcing extension of the set-theoretical universe.NEWLINENEWLINEMore specifically, the author defines the logic \({\mathcal L}^{1}_{\kappa}\) for any suitable cardinal \(\kappa\) playing the role of \(\aleph_0\) in first-order logic. It is proved that this logic is nice and satisfies the following: (a) A downward Löwenheim-Skolem-Tarski property to \(\kappa\). (b) If the vocabulary has cardinality \(< \kappa\), then the number of sentences is \(\kappa\). (c) A weak substitute for compactness. (d) The theory of a product of two models depends just on the theories of the two models. (e) Interpolation. Furthermore the logic has reasonable characterizations; it is placed between classical infinitary logics and has both maximality and minimality properties. The definition of the logic does not sound like that of a logic but the author intends to give a presentation closer to traditional definitions of a logic in a subsequent paper continuing the present one.NEWLINENEWLINEThe definition of the new logic takes places in the second section of the paper. It is based on a generalization of the Ehrenfeucht-Fraïssé game, ``allowing rescheduling of debts'' in the author's terms. Since this does not give an equivalence relation, the author has to close the induced relation to some equivalence relation in order to obtain classes of models of a sentence. The deeper properties of the logic, such as interpolation, are proved in the third section. In the fourth and final section the author discusses how strong the logic is and deals with sums and products of models.