"Each nation can do well, as in the life of individuals, only if things go equally well in all of her neighboring nations; the life of sciences is similar to that of states whose interest demands that everything be in order within individual states as properly as their relations be in good order among themselves. Understanding this correctly, the most important carriers of mathematical thoughts have always shown great interest in the law and order in neighboring sciences and, above all (...) for the benefit of mathematics, have cultivated the relations to the neighboring sciences, in particular to physics and epistemology. The essence of these relations and the ground of their fertility will be made most distinct, I believe, if I sketch to you that general method of inquiry which appears to grow more and more significant in modern mathematics; the axiomatic method, I mean." (David Hilbert, 1917). (shrink)

In this note, a fully modal proof is given of some conservation results proved in a previous paper by arithmetic means. The proof is based on the extendability of Kripke models.