Let $\lambda$ be a singular cardinal. Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$?


(Note that even without the size requirement this implies a failure of the Jensen covering property, so such a forcing does not necessarily exist.)



collapsing successor of singular
The answer is no and it follows easily from the following theorem:



> 
> **Theorem.** Suppose $\kappa$ is a regular uncountable cardinal and $|P|\leq \kappa.$ Then $\Vdash\_P cf(\kappa)=|\kappa|.$
> 
> 
> 


For a proof of the theorem see
[Singularizing forcing of "small" cardinality?](https://mathoverflow.net/questions/142086/singularizing-forcing-of-small-cardinality)


