Full Idea
We cannot explicitly define one-one correspondence from the sets to the ordinals (because there is no explicit well-ordering of R). Nevertheless, the Axiom of Choice guarantees that a one-one correspondence does exist, even if we cannot define it.
Clarification
'R' is the real numbers
Gist of Idea
The Axiom of Choice guarantees a one-one correspondence from sets to ordinals
Source
Keith Hossack (Plurals and Complexes [2000], 10)
Book Reference
-: 'British Soc for the Philosophy of Science' [-], p.436