Model theory for \(L_{\infty \omega _ 1}\) (Q762061)

The fact that finiteness can be expressed in the infinitary language \(L_{\infty,\omega}\) is a barrier to a general compactness result for \(L_{\infty,\omega}\). However Barwise and Kreisel showed how to obtain a useful version of ''compactness'' by looking at certain countable fragments of \(L_{\infty,\omega}\). Similarly, the fact that well-foundness is expressible in \(L_{\infty,\omega_ 1}\) has been a stumbling block to compactness results for (uncountable) fragments of \(L_{\infty,\omega_ 1}\). In the paper it is shown that provided one restricts oneself to ''effectively scattered'' theories of \(L_{\infty,\omega_ 1}\), a compactness theorem holds. The paper ends with some applications. However it is still too early to see if this compactness theorem will be as useful as Barwise's compactness theorem.