[jhghikvhu]

> But now let's divide each element of E by 2 to produce a set D.

You are assuming that this doesn't change the size and certainly that's how the normal notion of size works. But the question is whether we can create any order relationship on the sets with the desired properties.

The properties he mentioned defines a partial order and partial orders can be extended to total orders (given axiom of choice). So it is in fact possible.

[pfdietz]

A partial order of the elements of some set can be extended to a total order.

But a partial order of all sets? How is this function "size" even defined? Functions don't have a domain of all sets in conventional set theory.