TITLE: Can $T$, $T+A$, and $T+\neg A$ all have different consistency strengths?
QUESTION [8 upvotes]: Let $T$ be a consistent theory, and let $A$ be a statement in the same language. Consider the three theories

$T$
$T+A$
$T+\neg A$

Is it possible for them to be pairwise distinct in consistency strength?
As a follow-up, is it possible for $T+A$ and $T+\neg A$ to be incomparable in consistency strength? (Clearly they are both stronger than $T$, so a fortiori this would mean that the three consistency strengths are distinct.)
I'm primarily interested in theories in classical, finitary first-order logic, but if it makes a difference to consider other logics, I'd find that interesting.
If $T+A$ is inconsistent, then $\neg A$ is provable in $T$, so $T+\neg A$ and $T$ are certainly equiconsistent; similarly if $T+\neg A$ is inconsistent, then $T$ and $T+A$ are equiconsistent. So the question is only interesting if all theories involved are consistent.
This question is motivated by the fact that typically if $T$ is ZFC and $A$ is a large cardinal hypothesis, then $T+\neg A$ is modeled by a substructure of any model of $T$ (by "chopping the universe off" at an inaccessible), so that $T$ and $T+\neg A$ are equiconsistent.

REPLY [3 votes]: You can construct such sentences via fixed point theorem; viz. there exists $\phi$ such that both $Con(ZFC + \phi)$, $Con(ZFC + \neg \phi)$ are unprovable in $ZFC + Con(ZFC)$. Such a $\phi$ is what is called a double jump sentence. A reference for this is P. Lindstrom, Aspects of Incompleteness, 2003.