Difference between revisions of "Nonstandard satisfaction classes."

From Peano's Parlour
Jump to: navigation, search
(Created page with " == 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 eit...")
 
Line 1: Line 1:
 
 
== Fullness of M ==
 
== Fullness of M ==
  
Line 6: Line 5:
 
Fullness of MFull(M)} is the set of those eM  M for which M  has an e-full inductive partial satisfaction class.
 
Fullness of MFull(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 $\{\rm Full}(M)containsanelementgreaterthanalldefinableelementsofM,then{\rm Full}(M)=M$.\end{enumerate} \end{prop}
+
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 MTFull(M) contains no definable nonstandard elements.  
 
Kaufmann and Schmerl showed that there are  completions T of PA, such that for every MTFull(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 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 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.