Modern set theories like ZFC and NBG handle the totality of cardinals via the proper class Ord without invoking a non-arithmetical 'absolute infinity' as a distinct metaphysical category.
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(Classical set theory as a foundation for mathematics)
The axiom system ZF plus the axiom of choice (AC).
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.
non-arithmetical(Complexity classification of dependence logic validity)
Not decidable within the arithmetical hierarchy
ord(Gentzen's consistency proof for PA)
A primitive recursive assignment of ordinal representations (for ordinals less than epsilon_0) to proofs, used to measure proof complexity in Gentzen's reduction.
proper class(If the iterative process never terminated, the union of all generated chains would form a proper class, contradicting the assumption that P is a set.)
A collection of sets that is too large to itself be a set in the relevant set-theoretic framework.