# Natural theories for the failure of gap-1 transfer principle

The gap-1 transfer principle says that the transfer principle $(\kappa^+, \kappa) \to (\lambda^+, \lambda)$ holds for all infinite cardinals $\kappa, \lambda.$

It is known that for some sentence $\phi,$ $\phi$ has a $(\kappa^+, \kappa)$-model iff there exists a special $\kappa^+$-Aronszajn tree, and by a result of Mitchell, we can get the failure of gap-1 transfer $(\aleph_1, \aleph_0) \to (\aleph_2, \aleph_1)$ using a Mahlo cardinal (by producing a model in which there are no special $\aleph_2$-Aronszajn trees).

In his thesis, Rasch, introduced another theory, and proved the consistency of the failure of gap-1 using just an inaccessible cardinal. However his model is essentially Mitchell's model, but constructed using an inaccessible instead of a Mahlo cardinal.

I am wondering what other natural theories one can consider to obtain the failure of gap-1 transfer principle $(\aleph_1, \aleph_0) \to (\aleph_2, \aleph_1)$?. In particular I'm interested to obtain such a failure using models different from Mitchell's forcing (and of course a different theory than those stated above).

Giving references is also appreciated.

• I was looking for such examples too, but the only one I found is the one you mentioned. It is proved in Chang and Keisler. Will an $L_{\omega_1,\omega}$-sentence with models of type $(\aleph_1,\aleph_0)$, but no models of type $(\aleph_2,\aleph_1)$ work for your purposes? – Ioannis Souldatos Mar 22 '17 at 15:09
• @IoannisSouldatos Though such an example does not the job for me, but I'm interested to see it. – Mohammad Golshani Apr 3 '17 at 4:31