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\(L_{\infty \omega_ 1}\)-elementary equivalence of \(\omega_ 1\)-like models of PA. (English) Zbl 0545.03018

Models of PA are called similar if they are elementarily equivalent and have the same standard systems. It is shown that recursively saturated, \(\omega_ 1\)-like models of PA are \(L_{\infty \omega_ 1}\)- elementarily equivalent iff they are similar. Also, a construction of continuum many pairwise nonisomorphic recursively saturated, \(\omega_ 1\)-like similar models is given. Further results in the direction of the paper, including stronger versions of the results stated above can be found in the author’s paper ”Recursively saturated \(\omega_ 1\)-like models of arithmetic” [Notre Dame J. Formal Logic (to appear)].

MSC:

03C62 Models of arithmetic and set theory
03C52 Properties of classes of models
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