Assuming w is the only measurable cardinal, we prove: Let ˜ be an equivalence relation such that ˜ = =l for some logic L < L* satisfying Robinson’s consistency theorem (with L* arbitrary); then there exists a strongest logic L+ < L* such that ˜ = =l+; in addition, L+ is countably compact if. Let ˜ be an equivalence relation such that ˜ = =tp for some logic L° satisfying Robinson’s consistency theorem and whose sentences of any type r are (up to equivalence) equinumerous with some cardinal kt; then L° is the unique logic L such that ˜ = =l\ furthermore, L° is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations ˜ which are equal to =l for some compact logic L obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.
Duality between logics and equivalence relations / D. MUNDICI. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 270:(1982), pp. 111-129. [10.2307/1999763]
Duality between logics and equivalence relations
MUNDICI, DANIELE
1982
Abstract
Assuming w is the only measurable cardinal, we prove: Let ˜ be an equivalence relation such that ˜ = =l for some logic L < L* satisfying Robinson’s consistency theorem (with L* arbitrary); then there exists a strongest logic L+ < L* such that ˜ = =l+; in addition, L+ is countably compact if. Let ˜ be an equivalence relation such that ˜ = =tp for some logic L° satisfying Robinson’s consistency theorem and whose sentences of any type r are (up to equivalence) equinumerous with some cardinal kt; then L° is the unique logic L such that ˜ = =l\ furthermore, L° is compact and obeys Craig’s interpolation theorem. We finally give an algebraic characterization of those equivalence relations ˜ which are equal to =l for some compact logic L obeying Craig’s interpolation theorem and whose sentences are equinumerous with some cardinal.I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.