Zermelo's 1908 construction is now meant to define a ‘remainder set’ directly without detour through some ≺; the resultant inclusion ordering is then ‘mirrored’ on M. The key thing is to show that the chain of subsets of M picked out really matches M itself. But if there were some element a ∈ M which did not correspond to a remainder ℜ≼(a), then it must be possible to use the choice function to ‘squeeze’ another remainder into the chain, which would contradict the assumption that all the sets with the appropriate definition are already in the chain.[29] We have spoken of functions and relations here. But in fact Zermelo avoids such talk. He defines M as being ‘well-ordered’ when each element in M ‘corresponds’ uniquely to such a ‘remainder’ (Zermelo 1908a: 111). This shows, says Zermelo, that the theory of well-ordering rests ‘exclusively upon the elementary notions of set theory’, and that ‘the uninformed are only too prone to look for some mystical meaning behind Cantor's relation a ≺ b’ (Zermelo 1908a). 