# Is not SH + not CH consistent?

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?

• HS or SH? And what exactly do you mean by that? Suslin Hypothesis? In that case, taking from Wikipedia's enty on Suslin's problem: "The Suslin hypothesis is independent of ZFC, and is independent of both the generalized continuum hypothesis (proved by Ronald Jensen[3]) and of the negation of the continuum hypothesis." – Asaf Karagila Aug 9 '14 at 9:53
• Sorry, it was SH. I am aware that Jensen obtained a model for GCH + SH. I suppose that Wikipedia refers to this fact. I have studied a modern proof of this result (a variant of Shelah's one) based on a countable support iteration of proper forcing posets, but I do not know any model of $\lnot$SH + $\lnot$CH. The Souslin trees are usually constructed with $\Diamond$, which implies CH. – Carlos Aug 9 '14 at 10:10

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...)

1. Jech, Set Theory, 3rd Millennium ed., Theorem 28.12, p.563

2. Saharon Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math. 48 (1984), no. 1, 1--47.

Two more solutions.

1. Start with a model with a Suslin tree $(T,\leq)$ in it (force it or use L). Let $(P,\leq)$ be the Cohen forcing adding any number of reals. We claim that $(T,\leq)$ remains Suslin after forcing with $(P,\leq)$. Assume that $p$ forces that $A$ is an uncountable antichain in $T$. There are conditions $p_\alpha\leq p$ and distinct elements $t_\alpha\in T$ such that $p_\alpha$ forces that $t_\alpha\in A$. An easy argument (using the delta system lemma) shows that there is an uncountable set $Z$ such that any finite subfamily of $\{p_\alpha:\alpha\in Z\}$ has a common lower bound. Then $\{t_\alpha:\alpha\in Z\}$ is an antichain, as if $\alpha\neq\beta$ are in $Z$, then some $p'\leq p_\alpha,p_\beta$ forces that $t_\alpha$, $t_\beta$ are both in $A$. Contradiction to the fact that $T$ is Suslin in the ground model.

2. In Jech's above mentioned book, there is a forcing adding a Suslin tree with finite conditions (in the exercises). Force with it over a model of non-CH.

In K. Devlin: $\aleph_1$ trees, Annals of Math. Logic, 13(1978), 267-330, all combinations of CH, ST (there exists a Suslin tree) and KT (there exists a Kurepa tree) and their negations are considered.