Nonstandard satisfaction classes.

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Fullness of M

A satisfaction class for a model M is e-full, if S decides all If for every Σe sentence (in the sense of M) φ with parameters either φS or ¬φS.

Fullness of M, Full(M)} is the set of those eM M for which M has an e-full inductive partial satisfaction class.

It is easy to see that Full(M) is a cut of M and, if Full(M)>ω then M is recursively saturated. Also if M is countable and \{\rm Full}(M) contains an element greater than all definable elements of M, then Full(M)=M.\end{enumerate} \end{prop}

Kaufmann and Schmerl showed that there are completions T of PA, such that for every MT, Full(M) contains no definable nonstandard elements.

Problem: Suppose Full(M)=M, does M have a full inductive satisfaction class?

Reference: Kaufmann, Matt; Schmerl, James H. Remarks on weak notions of saturation in models of Peano arithmetic. J. Symbolic Logic 52 (1987), no. 1, 129–148.