"... They dined on mince, and slices of quince."

Plural nomenclature comes to us as early in life as poems like The Owl and the Pussy-Cat. We may go on to ask how many were the slices of quince on the table. Were they (the slices of quince) long or were they short, etc. Let us take as a starting point that the primary formal name for a class is a plural name like the slices of quince. We may then ask whether the class follows the same logical rules as the set of mathematical set theory. The answer is "no".

  Courtesy one of the more beautiful items on YouTube. Words by Emmerson, Lake and Palmer.
 ... improper class and the energy of our gravitational field ...
Purging the Zermelo canon involves refreshing our acquaintance with plural nomenclature as found in the law of contest. The law of contest arises as a preferred alternative for when the law of civil emergency and the law of reserved lands are inapplicable. Through finely crafted general rules or through instruments, the law of contest is designed to elevate disputes in the learnedness of the parties perceptions and conceptions.

In the language of the law of contest, we may address the question of how many slices of quince were on the table. In this mode we are, as it were, addressing the sum. In another matter, if we bought three horses from Brown on Friday and they all died over the weekend, how many could be left? Here also, we could be addressing the sum. But we may also address the class. We may indeed ask what is referred to by the horses we bought from Brown except the ones now dead. On the assumption that they're all dead, this name refers to zero, also known as zero the real number. In this degenerate case, the class is no more than a pure number. However zero is not included in every class. As the null set is a subset of every set, we may start to become confused if we identify classes as sets.

Further discrepancy arises in connection with the dimension of member sort. Class inclusion has its transitivity rule held up by a condition with no parallel in the theory of sets. This is connected with the idea that certain facts of proportion are intrinsic to (say) unicorn nature. A unicorn with the horn at the back rather than at the front must be called "odd" as in "odd unicorn" because members have their sort of thing somewhat defined by learned agreement. Such agreements must for all practical purposes exhaust the logical universe. Hence we have a universal distinction between nations and their parts, between elephants and their parts, between puzzlers and their parts. Classes compared may break lines of sort and transitivity of inclusion must be qualified with a reference to sort.

As children acquire most of the framework for formal language by the time they are eleven or twelve, we may need to be careful about what we teach them in elementary mathematics lessons. In the old days, mathematics teachers were free to express their altruism. Representing their love for the language of the law of contest, class theory, rather than set theory, may have been the favoured vehicle. Do we now standardise out all of that, as to gather the world around the Zermelo canon? One hopes not.