If κ's position in the inaccessible hierarchy is model-relative, then 'κ cannot be the nth inaccessible' is not an absolute claim but a schema relativized to a background universe.
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(Large cardinal assumption in the structuralist account)
A cardinal number large enough to serve as the cardinality of the mereological atoms in the universe, enabling the structuralist construction of set theory.
model-relative(describing properties in logic and mathematics)
Dependent on which particular version or interpretation of a system you're looking at; something can be true in one model but not in another.
κ (kappa)(mathematical notation)
A symbol used in set theory to represent a large cardinal number (roughly: a really, really big number with special properties).