TITLE: (Non?)-linearity of the consistency strength ordering in ZF
QUESTION [11 upvotes]: Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all sentences $\sigma,\tau\in Sent$, $\sigma$ has less consistency strength than $\tau$ (or $\sigma\leq_{cons}\tau$) iff $cons(ZF+\tau)\rightarrow cons(ZF+\sigma)$. My main question (which I believe should have a negative answer) is the following:
$$\mbox{ Is } \leq_{cons}\mbox{ linear? If not is there a natural counterexample?}$$
In fact I have several similar questions concerning $\leq_{cons}$ and I would be glad if someone could provide me with some references. Thanks!

REPLY [6 votes]: Unlike - seemingly - all the answers above, and since strangely he seems to have missed this question (as not so often), Pr. J.D. Hamkins claims he has natural statements that contradict the pre-linearity of this pre-ordering!
He will talk about them on 25 January at 7pm UTC:
http://jdh.hamkins.org/natural-instances-of-nonlinearity-in-the-hierarchy-of-consistency-strength-uwm-logic-seminar-january-2021/