[FOM] Functions from sets to ordinals

[Joe Shipman]

The following functions from hereditarily finite sets to natural numbers have natural extensions to functions from arbitrary sets to ordinals:

1) cardinality 
2) rank
3) nim-value (the nim-value of a set is the first ordinal not the nim-value of any of its members)
4) cardinality of V_rank

These functions all have the property that if A is a subset of B, then f(A) <= f(B).

A fifth function satisfying this property is the Ackermann enumeration of HF sets:
4) f(A) is the sum over all x in A of 2^f(x).

The finite ordinals themselves map as follows in these cases:
1) 0,1,2,3,...
2) 0,1,2,3,...
3) 0,1,2,3,...
4) 0,1,2,4,16,65536,...
5) 0,1,3,11,2059,...

How far can this last function be generalized? If you have a global choice function or a global well-ordering given in a class theory like NBG, can you define a global well-ordering which reduces to 5) for HF sets and has the monotonicity property?

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