Difference between revisions of "Cuts in recursively saturated models"

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This problem is listed in Kossak Roman; Schmerl, James H.
 
This problem is listed in Kossak Roman; Schmerl, James H.
 
The structure of models of Peano arithmetic, but, unfortunately, with many typos.
 
The structure of models of Peano arithmetic, but, unfortunately, with many typos.
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== Elementarily equivalent nonisomorphic pairs? ==
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Let MPA be countable and recursively saturated and let K and K be short elementary cuts of M such that (M,K)(M,K). Are (M,K) and (M,K) isomorphic?
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Reference: Kossak Roman; Schmerl, James H. ''On Cofinal Submodels and Elementary Interstices'', Notre Dame Journal of Formal Logic, 53, No 3, 2012, pp. 267-287.

Revision as of 09:32, 20 January 2013

Free cuts

A cut I in a model MPA is free if for all a,bI 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 MPA 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 YM is coded in N, then (M,Y)(M,X)?


The answer if `yes' is either (M,X)PA or Th(M,X)SSy(M).

This problem is listed in Kossak Roman; Schmerl, James H. The structure of models of Peano arithmetic, but, unfortunately, with many typos.


Elementarily equivalent nonisomorphic pairs?

Let MPA be countable and recursively saturated and let K and K be short elementary cuts of M such that (M,K)(M,K). Are (M,K) and (M,K) isomorphic?

Reference: Kossak Roman; Schmerl, James H. On Cofinal Submodels and Elementary Interstices, Notre Dame Journal of Formal Logic, 53, No 3, 2012, pp. 267-287.