Let Robinson's consistency theorem hold in logic L: then L will satisfy all the usual interpolation and definability properties, together with coutable compactness, provided L is reasonably small. The latter assumption can be weakened ro removed by using special set-theoretical assumptions. Thus, if Robinson's consistency theorem holds in L, then (i) L is countably compact if its Löwenheim number is < μ0 = the smallest uncountable measurable cardinal; (ii) if ω is the only measurable cardinal, L is countably compact, or the theories of L characterize every structure up to isomorphism. As a corollary, a partial answer is given to H. Friedman's third problem, by proving that no logic L strictly between L∞ω and L∞∞ satisfies interpolation (or Robinson's consistency), unless K-elementary equivalence coincides with isomorphism.

Compactness, interpolation and Friedman's third problem / D. MUNDICI. - In: ANNALS OF MATHEMATICAL LOGIC. - ISSN 0003-4843. - STAMPA. - 22:(1982), pp. 197-211. [10.1016/0003-4843(82)90021-3]

Compactness, interpolation and Friedman's third problem

MUNDICI, DANIELE
1982

Abstract

Let Robinson's consistency theorem hold in logic L: then L will satisfy all the usual interpolation and definability properties, together with coutable compactness, provided L is reasonably small. The latter assumption can be weakened ro removed by using special set-theoretical assumptions. Thus, if Robinson's consistency theorem holds in L, then (i) L is countably compact if its Löwenheim number is < μ0 = the smallest uncountable measurable cardinal; (ii) if ω is the only measurable cardinal, L is countably compact, or the theories of L characterize every structure up to isomorphism. As a corollary, a partial answer is given to H. Friedman's third problem, by proving that no logic L strictly between L∞ω and L∞∞ satisfies interpolation (or Robinson's consistency), unless K-elementary equivalence coincides with isomorphism.
1982
22
197
211
D. MUNDICI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/308866
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