If class bijections require definable functions, the step from equinumerosity with the ordinals to a global well-ordering presupposes definable choice functions, making the argument question-begging.

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choice functions(Formal semantics of indefinites): Functions that select a member from a set, used as a semantic mechanism to account for the scopal behavior of indefinites without syntactic movement
class bijections(as used in set theory and mathematics): A one-to-one matching between two groups of things where each item in one group pairs with exactly one item in the other group with no leftovers.
definable functions(as used in mathematical logic): Rules for matching or transforming things that can be precisely described or written down in a clear, explicit way.
equinumerosity(as used in set theory): The property of two groups having the same size or number of items, even if the items themselves are different.
global well-ordering(Derived from a class bijection between the universe and the class of von Neumann ordinals): A well-ordering of the entire universe of sets, whose existence implies the Axiom of Choice
ordinals(Proof-theoretic treatment of ordinals, distinct from but related to set-theoretic ordinals): A central concept in both set theory and proof theory, used by Gentzen to assign measures to proofs in order to demonstrate consistency of PA via well-foundedness
question-begging(Epistemology, anti-skeptical argumentation): A charge leveled against anti-skeptical arguments that assume what they set out to prove, particularly in Putnamian externalist arguments

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