The claim holds only within ZFC set theory; in alternative foundations like NFU or Aczel's non-well-founded set theory, the relationship between ordinal and cardinal structure is not preserved in the same way.
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(describing whether ordinal and cardinal structure maintain their relationships)
In this context, it means the relationships between two ideas work the same way or stay consistent.
Set theory(as used in mathematics)
A branch of mathematics that studies collections of objects (called 'sets') and the rules for how they relate to each other.
ZFC set theory(as the primary formal system being discussed)
A standard system of rules for organizing and working with collections of things (called sets) in mathematics; most mathematicians use this system as their foundation.
cardinal(in mathematics and set theory)
A number that represents the size of a set (how many things are in it), as opposed to the order they're in.