|
|
| Line 1: |
Line 1: |
| − | $X\subseteq M\models PA$ 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.
| |
| | | | |
| − | Using the fact that every model of $PA$ has a conservative elementary end extension <cite> gaifman1976:models </cite>, Schmerl <cite> schmerl1981:recursively </cite> proved that 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 <cite> kaufmann1977:arather </cite>, assuming $\lozenge$, proves that there are recursively saturated $\aleph_1$-like rather classless models. Later Shelah showed that $\lozenge$ can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in <cite> hodges1985:building </cite>: Prove the existence of rather classless recursively saturated models of $PA$ in cardinality $\aleph_1$ without assuming diamond at any stage of the proof.
| |
| − |
| |
| − |
| |
| − | {{References}}
| |
