Is there a known and strong reason why the large cardinal hierarchy is almost linear w.r.t. the consistency strength?
2026-03-25 16:46:28.1774457188
Consistency strength of the Large cardinals is almost linear
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This is commonly regarded as open (that is, nobody truly knows why this is). Hugh Woodin's $\Omega$-logic program (an infinitary logic with important properties) hopes to solve this and perhaps even give a platonic "answer" to the Continuum Hypothesis.
You can read about in the links mentioned on this page, however it is still a controversial matter as to whether it resolves anything. The essential idea is that there is a notion called $\Omega$-provability which allows us to create a quite general definition of a large-cardinal axiom. It then shows that there is a sort of proof-hierarchy in set theory so that the large cardinals each correspond nicely with each rank...
Apologies if this answer is quite vague - this is quite cutting-edge stuff!