Geodesic
Construction of Satisfaction Classes for Nonstandard Models
Canadian mathematical bulletin, Tome 24 (1981) no. 3, pp. 283-293

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Given a resplendent model for Peano arithmetic there exists a full satisfaction class over , i.e. an assignment of truth-values, to all closed formulas in the sense of with parameters from , which satisfies the usual semantic rules. The construction is based on the consistency of an appropriate system of -logic which is proved by an analysis of standard approximations of nonstandard formulas. 
Kotlarski, H.; Krajewski, S.; Lachlan, A. H. Construction of Satisfaction Classes for Nonstandard Models. Canadian mathematical bulletin, Tome 24 (1981) no. 3, pp. 283-293. doi: 10.4153/CMB-1981-045-3
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     title = {Construction of {Satisfaction} {Classes} for {Nonstandard} {Models}},
     journal = {Canadian mathematical bulletin},
     pages = {283--293},
     year = {1981},
     volume = {24},
     number = {3},
     doi = {10.4153/CMB-1981-045-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1981-045-3/}
}
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