Difference between revisions of "Cuts in recursively saturated models"
(Created page with "== Free cuts == A cut I in a model M⊨PA is ''free'' if for all a,b∈I if (M,a)≡(M,b), then (M,I,a)≡(M,I,b). There are free elementary cuts in ev...") |
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− | Kossak, Roman Four problems concerning recursively saturated models of arithmetic. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519–530. | + | Kossak, Roman ''Four problems concerning recursively saturated models of arithmetic''. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519–530. |
− | Kaye, Richard; Wong, Tin Lok Truth in generic cuts. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005. | + | Kaye, Richard; Wong, Tin Lok ''Truth in generic cuts''. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005. |
Revision as of 10:22, 18 January 2013
Free cuts
A cut I in a model M⊨PA is free if for all a,b∈I if (M,a)≡(M,b), then (M,I,a)≡(M,I,b). There are free elementary cuts in every countable recursively saturated model of PA, and generic cuts of Kaye and Tin Lok Wong are free.
Problem: Let M⊨PA be countable and recursively saturated. Does M have a free elementary cut I such that the pair (M,I) is recursively saturated?
References:
Kossak, Roman Four problems concerning recursively saturated models of arithmetic. Special Issue: Models of arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519–530.
Kaye, Richard; Wong, Tin Lok Truth in generic cuts. Ann. Pure Appl. Logic 161 (2010), no. 8, 987–1005.
Omitting theories of subsets
Suppose M is countable recursively saturated and X is an undefinable subset of M. Is there a countable recursively saturated N such that N is an elementary end extension of M, and if Y⊆M is coded in N, then (M,Y) \not\equiv (M,X)?
The answer if `yes' is either (M,X)\not\models PA^* or {\rm Th}(M,X)\notin {\rm SSy}(M).
This problem is listed in Kossak Roman; Schmerl, James H. The structure of models of Peano arithmetic, but, unfortunately, with many typos.