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“they don't work at all when one wants to look at "well-ordered sets and their structure-preserving maps", that is, monotone maps: there are two monotone maps from (1, <) to (2, <), but only one from {{}} to {{{}}}, so there's a lot of information loss” - that’s interesting - but why is it better that we have 2 monotone maps instead of one? You’re saying the zermelo ordinals are perfectly adequate as a “rank function”, but the VN-ordinals express more - then I wonder why we can’t consider Z-ordinals as canonical representation of natural numbers, and VN-ordinals of ordered sets - keep both
– Julius Hamilton
Commented Apr 13 at 0:19
there are two monotone maps from any ordered 1-element to any ordered 2-element set, it's not a question of being "better" or anything. regarding the ranking function: zermelo's don't really work for transfinite rank (that's the content of the edit). in any case, we want to speak of/work with functions, and zermelo's ordinals are all singletons, so completely inadequate
– ac15
Commented Apr 13 at 0:46

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