Difference between revisions of "Rather classless models."
(Created page with " X⊆M⊨PA is a ''class'' if for all a∈M, {x∈X:x<a} is definable (coded) in M. M is ''rather classless'' if each class of M is definable. Since...") |
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X⊆M⊨PA is a ''class'' if for all a∈M, {x∈X:x<a} is definable (coded) in M. | X⊆M⊨PA is a ''class'' if for all a∈M, {x∈X:x<a} is definable (coded) in M. | ||
| − | M is ''rather classless'' if each class of M is definable. | + | M is ''rather classless'' if each class of M is definable. |
| − | Since every model of PA has a conservative elementary end extension, for each cardinal κ such that \cf(κ)>ℵ0, there are κ-like rather classless models of PA. A model is ''κ-like'' is it is of cardinality κ and each of its proper initial segments is of smaller cardinality. | + | Since every model of PA has a conservative elementary end extension, for each cardinal κ such that ${\rm cf}(\kappa)>\aleph_0,thereare\kappa−likeratherclasslessmodelsofPA.Amodelis″\kappa-like'' is it is of cardinality \kappa$ and each of its proper initial segments is of smaller cardinality. |
Kaufmann, assuming \lozenge, proves that there are recursively saturated \aleph_1-like rather classless models. Kaufmann, Matt, ''A rather classless model''. Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333. Later Shelah showed that \lozenge can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in | Kaufmann, assuming \lozenge, proves that there are recursively saturated \aleph_1-like rather classless models. Kaufmann, Matt, ''A rather classless model''. Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333. Later Shelah showed that \lozenge can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in | ||
Hodges, Wilfrid, ''Building models by games''. London Mathematical Society Student Texts, 2. Cambridge University Press, Cambridge, 1985: Prove the existence of rather classless recursively saturated | Hodges, Wilfrid, ''Building models by games''. London Mathematical Society Student Texts, 2. Cambridge University Press, Cambridge, 1985: Prove the existence of rather classless recursively saturated | ||
models of PA in cardinality \aleph_1 without assuming diamond at any stage of the proof. | models of PA in cardinality \aleph_1 without assuming diamond at any stage of the proof. | ||
Revision as of 16:04, 17 January 2013
X\subseteq M\models PA is a class if for all a\in M, \{x\in X: x<a\} is definable (coded) in M. M is rather classless if each class of M is definable.
Since every model of PA has a conservative elementary end extension, for each cardinal \kappa such that {\rm cf}(\kappa)>\aleph_0, there are \kappa-like rather classless models of PA. A model is \kappa-like is it is of cardinality \kappa and each of its proper initial segments is of smaller cardinality.
Kaufmann, assuming \lozenge, proves that there are recursively saturated \aleph_1-like rather classless models. Kaufmann, Matt, A rather classless model. Proc. Amer. Math. Soc. 62 (1977), no. 2, 330–333. Later Shelah showed that \lozenge can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in Hodges, Wilfrid, Building models by games. London Mathematical Society Student Texts, 2. Cambridge University Press, Cambridge, 1985: Prove the existence of rather classless recursively saturated models of PA in cardinality \aleph_1 without assuming diamond at any stage of the proof.
