I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?



Is not SH + not CH consistent?
Recall that adding a single Cohen real adds a Suslin tree, and does not change the value of the continuum [1] [2]. So by taking any model of $\lnot\sf CH$ and adding a single Cohen real, we have a Suslin tree, and therefore $\lnot\sf SH$ as well.


(This makes the Solovay-Tennenbaum theorem all the more magical, since we add so many Suslin trees in the iteration, but we still manage to kill them all eventually...)




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1. Jech, **Set Theory, 3rd Millennium ed.**, Theorem 28.12, p.563
2. *Saharon Shelah*, [**Can you take Solovay’s inaccessible away?**](http://dx.doi.org/10.1007/BF02760522), *Israel J. Math.* **48** (1984), no. 1, 1--47.


