Ordering of Large Cardinal Axioms One of the unexplained phenomena about large cardinal axioms is that they tend to be in a linear order (in terms of consistency strength). So it brings to mind other natural questions about this "order".
So my question is -
For any two large cardinal axioms $A1$, $A2$ such that $ZFC + A1$ proves $ZFC + A2$ is consistent, can we define a large cardinal axiom $A3$ such that $ZFC + A1$ proves $ZFC + A3$ is consistent, and $ZFC + A3$ proves $ZFC + A2$ is consistent? 
Meaning, is the order known\thought to be densely ordered (assuming the ordering itself is not some random phenomena)?
Alternatively, can we build an example (even a highly contrived one) of $A1$, $A2$, where any axiom we add to $ZFC$ is either stronger than $ZFC + A1$, weaker than $ZFC + A2$, or equiconsistent to one of them (meaning the ordering is not dense)?
P.S - I know there is no agreed upon definition of "large cardinal axiom", but if we discuss their linear order we should be able to ask questions like this as well.
 A: If by large cardinal axioms you mean axioms which directly allude to cardinals, then the answer is negative.
Namely, if a large cardinal axiom is something like "There exists a cardinal $\kappa$ with such and such properties" and these properties imply certain amount of inaccessibility, in which case we rule out things like "$\sf ZFC$ is consistent" or "$0^\#$ exists", then the answer is negative.
To see that, simply note that there will always be a smallest large cardinal. If you want it to be such that $V_\kappa\models\sf ZFC$, then the least worldly cardinal is your smallest large cardinal. Of course this is debatable, and we don't have a mathematical definition of what is a large cardinal axiom. But much like pornography, "we know it when we see it".
So if we accept that worldly cardinals are the smallest large cardinals, then "There are two worldly cardinals" proves the consistency of "There is a worldly cardinal", but there are no intermediate notions to be found.
Of course you might argue that $\sf ZFC$ with "the theory $\sf ZFC+\exists \kappa$ worldly is consistent" proves the consistency of the existence of worldly cardinals, but now this becomes a question of whether or not this is a large cardinal axiom. Some people might argue that it is, others might argue that it's not.
For what it's worth, statement about "this theory is consistent" are arithmetic, so if we only assume $\sf PA$ with "this theory is consistent" then we can prove the consistency of said theory. But this is really not what we mean when we talk about large cardinals, is it now?
