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Mike, you are intelligent:
How-about try figure this out:
Dr. D. has a set of: arbitrary sized groups each of arbitrary numbers. He calls these "observations".
He wants to put them in a specific sequence from first to last group. He will do this by naming each group first, so he knows which group he is putting where in the sequence. He calls that arbitrary naming "time".
He wants to look at the minimum logical structure that a specific order would involve in re-ordering his arbitrary order. A specific order he supposes would be determined by a rule.
He wants the members of his set of "observations" (groups of numbers) to be free to be smaller or larger than each other or to be "identical" to each other. (In actual fact, if something is "plural", it is already a series of unique items, or you couldn't talk of "many" at all).
He adds so-called "unknown data" to each observation to allow for the possibility that some are smaller than others or some look the same as others within the limits of his view. But he overlooks? that his so-called "unknown data" must be complementary to what is in his sets.
E.g. if you have two observations: (5,5,5,5) and (5,5,5,5) you might add a 6 to the first one so you get (5,5,5,5,6). But if you already had somewhere a (5,5,5,5,6) then you have just gone and muddled your supposedly newly-uniquely identified observation with one of your others.
So each bit of added "unknown data" is actually going to have to be closely defined by the facts of what items are in the other observations and what items are added to them.
So the whole thing has to be self-referent through and through; the "unknown data" has to be very-well known within itself and its associations with the groups of numbers.
So his supposed imaginary "unknown data" has to be very carefully constructed and well known to his numbers and his groups of numbers to give the uniqueness he wants for each group (each "observation").
Now he "adds additional unknown data" such that even after removing an item, each set (observation) will be "unique". (Strictly of course: anything that is still "many" or "countable" is made of items "unique at some level" by definition; or you couldn't talk of "many" or use a plural).
His "additional unknown data" must also be carefully designed to be complementary to all the combinations and permutations already in his "groups of numbers with their attendent add-ons". Otherwise his new add-ons might re-muddle groups and make them look similar again.
So now he has got it so he can take any item away and still have all unique unmuddled observations (sets of numbers).
Now he has a situation where the series of removed items must always bind all the sets-of-numbers in a unique collection.
Of course any rule that required the sets of numbers to be in a special order also binds them together in at least one of two possible collections of optional orders.
Any rule must involve either a unique order or many possible orders but leaving at least one order free.
So any rule might be mapped by his uniquely defined order, or a collection of such orders.
I do not think he has thought it out fully. It can be very confusing.
-dolphin
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