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! 



(Non?)-linearity of the consistency strength ordering in ZF
Marios, this is indeed a fascinating topic. 


The consistency strength hierarchy is *not* linearly ordered. One can produce counterexamples by variants of Gödel sentences or of [Rosser sentences](http://en.wikipedia.org/wiki/Rosser%2527s_trick). It is actually an interesting exercise to produce explicit examples of a pair of Rosser-like sentences $\phi,\psi$ both independent of ZFC, such that ZFC$+\phi$ and ZFC$+\psi$ are incomparable, by letting $\phi$ refer to the length of proofs of consistency of ZFC$+\psi$, and letting $\psi$ refer to the length of proofs of the consistency of ZFC$+\phi$. Explicit examples can be found, I believe, in Per Lindström's "Aspects of Incompleteness" (Lecture Notes in Logic 10).


Harvey Friedman has worked on this question, and expanded on the idea above. I believe it is his result that one can even embed a universal countable partial order in the consistency strength degrees or in the degrees of *interpretability*. I'm pretty sure he has commented on this in the [FOM](http://www.cs.nyu.edu/mailman/listinfo/fom/) list, but I don't seem to be able to find his relevant postings right now. (I once gave a talk on this years ago, but cannot find my slides either. Old age...)


On the other hand, everybody agrees that these examples are not *natural* (see for example, the beginning of the section "The Gödel hierarchy" in [this paper](http://www.math.psu.edu/simpson/papers/gh/) by Simpson, or page 54 of "[Fixing Frege](http://books.google.com/books?id=qqdXsato74QC&printsec=frontcover&dq=fixing+frege&hl=en&src=bmrr&ei=BR2QTbGGDZScsQO6q_CDCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCgQ6AEwAA#v=onepage&q&f=false)" by John P. Burgess). It is a remarkable empirical phenomenon that we indeed have comparability for natural theories. We expect this to always be the case, and a significant amount of work in inner model theory is guided by this belief. But although we have many striking partial results in this direction, it is not quite a theorem, in part because *natural* is not a formal concept. 


The standard way by which we establish comparability of statements is by using forcing and the technique of inner models to measure a given statement against the large cardinal hierarchy, identifying this way the *large cardinal companion* of the statement. Different statement can then be compared by comparing their companions. 


Hugh Woodin, in his [book](http://rads.stackoverflow.com/amzn/click/3110197022) on Pmax suggests a way of formalizing this observation, by explaining how to associate [Universally Baire sets](http://en.wikipedia.org/wiki/Universally_Baire_set) to theories, the point being that under appropriate determinacy assumptions, the Wadge degrees of any two such sets are comparable, and the resulting ordering coincides precisely with the interpretability or consistency strength orderings. (The universally Baire sets ultimately trace back to the complexity of certain iteration strategies.) This, of course, requires that we work under additional large cardinal assumptions beyond ZFC, so it only applies to theories of high consistency strength (past $\omega$ Woodin cardinals, for example). This remarkable approach has thus the drawback that it does not explain why we find the phenomenon in the low levels of the hierarchy, the ones that are precisely amenable to current inner model theoretic techniques.


At the moment there are a few notable examples of theories that we do not quite know how to fit into the linear ordering of consistency strength. This is most likely due to our present technical limitations. For example, *strongly compact cardinals* are expected to either coincide or sit just below supercompact cardinals consistency-wise. However, their behavior is rather atypical and they are not really as well understood as other (strong) large cardinals.


For a nice introduction to these ideas, examples, and a far reaching program that touchesof Woodin's recent work on "ultimate $L$", see these [slides](http://math.berkeley.edu/~steel/talks/phila2010.pdf) of a talk by John Steel at the Workshop on Set Theory and the Philosophy of Mathematics, Oct. 2010.


