Difference between revisions of "Nonstandard satisfaction classes."
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Fullness of M, Full(M)} is the set of those e∈M M for which M has an e-full inductive partial satisfaction class. | Fullness of M, Full(M)} is the set of those e∈M 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 $ | + | 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 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 M⊨T, Full(M) contains no definable nonstandard elements. | Kaufmann and Schmerl showed that there are completions T of PA, such that for every M⊨T, Full(M) contains no definable nonstandard elements. |
Revision as of 12:28, 18 January 2013
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 φ∈S or ¬φ∈S.
Fullness of M, Full(M)} is the set of those e∈M 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 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 M⊨T, 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.