A quick question about strengths of inaccessible cardinals While me and my friend were discussing about set theory and stuff, my friend happened to send me a picture of a bunch of inaccessible cardinal arranged in a linear order, I'll attach the picture.
My question is what does this linear order represent, the strength? Consistency? I'm very new to inaccessible cardinals which is what makes me have this doubt. The order start with Mahlo, so does that mean Mahlo is weaker than all cardinals mentioned above it? But Upon reading Woodin cardinal in the wiki, I read that since Woodin cardinal is preceded by a stationary set of measurable cardinals, it's a Mahlo cardinal, so does that make Mahlo bigger? I'm utterly confused here, acc to the list it should be arranged strength wise cuz rank into rank is like the biggest inaccessible if we don't count the cardinals like reinhard and berkly which are inconsistent with ZFC. SO Just wanna know how the list is arranged, explainations would be more than appreciated!!
Here's the image
 A: The order shown is consistency strength, so e.g. Woodin being higher than weakly compact means that the consistency of "ZFC + there exists a Woodin cardinal" implies the consistency of "ZFC + there exists a weakly compact cardinal". Most, but not all, of the inequalities are known to be strict (i.e. it's been proven that the converse implication is not provable in any reasonable base theory).
We can also consider ordering large cardinals in terms of "size", but, as your confusion indicates, we need to be careful what we mean. Saying "$A\ge B$" is never meant in the sense that all cardinals with property $A$ are greater than all cardinals with property $B$ (i.e. the large cardinals do not form "layers"). It generally means one of two things: 1) Every cardinal with property $A$ has property $B$ or 2) The least cardinal with property $A$ is greater or equal to the least cardinal with property $B$. (And note that 1 implies 2.)
It is usually, but not always the case that cardinals lower on the diagram are of smaller size, in this sense, than ones higher up. For instance, every supercompact is a measurable, every measurable is a weakly compact, every weakly compact is a Mahlo, and every Mahlo is an inaccessible. So instead of forming layers, you can imagine the inaccessibles going "all the way up" and then every so often one of the inaccessiables is also a Mahlo, and every so often one of the Mahlos is also weakly compact, and so on.
One counterexample that you can surely read on the wikipedia page for Woodin cardinals, is that, provided Woodin cardinals exist, the least Woodin cardinal is not weakly compact. On the other hand, as you mention, every Woodin cardinal is Mahlo. So even though Woodins are much larger than weakly compacts and Mahlos in terms of consistency strength, in terms of "size", they are in between. (Notion 1 of "size", not notion 2... the least Mahlo cardinal is much, much larger than the least weakly compact since as you note, it is preceded by a stationary set of measurables.)
A: There is a similar diagram in Kanamori "The Higher Infinite," page 472, that we can look at for guidance. It is arranged by order of "direct implication or relative consistency implication, often both", where a higher item implies lower items. For example, Exercise 26.10 is "Suppose that $\kappa$ is Woodin. Then $\kappa$ is regular and $\{\alpha<\kappa\mid\alpha\text{ is measurable}\}$ is stationary in $\kappa$, and so $\kappa$ is $\kappa$-Mahlo" and Proposition 26.12 reads "If $\kappa$ is superstrong, then $\kappa$ is Woodin and there is a normal ultrafilter $U$ over $\kappa$ such that $\{\alpha<\kappa\mid\alpha\text{ is Woodin}\}\in U$." (pages 360-361). So many of these theorems are of the form "if $\kappa$ is $\phi(\kappa)$ then it is also $\psi(\kappa)$" (direct implication) or "If $Con(ZFC+\exists\kappa(\phi(\kappa)))$, then $Con(ZFC+\exists\kappa(\psi(\kappa)))$."
I believe the line $V=L$ is the largest cardinals compatible with the hypothesis $V=L$ (though I'm not certain, as I'm not familiar with iterable cardinals myself), and you are correct that this chart only considers $ZFC$.
