Mathematical collections

Sets are usually thought of as containing some objects (i.e. nonempty sets), rather then not containing anything at all (the empty set); sets are considered as the (unstructured) containers, and the empty set merely represents one extreme of that concept; the other is infinity (or, infinities).

Being unordered means that sets are unstructured containers, almost dumb boxes, but, in fact, sets have a rather impressing internal mechanism that can smack multiplicity out of an object, returning it to its true self (to its form)

Forms were properties or essences of things, treated as non-material abstract, but substantial, entities. They were eternal, changeless, supremely real, and independent of ordinary objects that had their being and properties by 'participating' in them.

However, this is a more natural way in which these theories were developed. Sequences were used long before sets, but if we extract the most elementary properties of sets and sequences, we get the two most fundamental ones: order and uniqueness. Both properties of order and uniqueness are realized in terms of a collection and its elements.

A collection is said to be ordered if it admits some notion of ordering of elements; meaning that it maintains its elements in some kind of order, it recognizes some type of ordering and it imposes it on the elements.

The property of uniqueness is about the metaphysical nature of objects, and the notion that each object has a unique form (a true self), along with any number of instances, which are just copies based on the object's essential form (like a class and its instances in OOP, like Plato's ideals). Collections that don't recognize such nonsense,

admit that there can be multiple copies of the same object

have the notion of multiplicity

The property of order is related to the structure; assume a nonempty one to make some sense. And the property of uniqueness is a property indicating whether a collection admits multiple instances of the same element or not.

of elements in a

as it specifies whether a structure admits some kind of ordering of the element within it. Since sets represent an unordered structure that disallows

attibute, we have to recognize a collection that is then

Elementary properties of collections:

• uniqueness (multiplicity)

• order (ordering)

Mathematical collections:

• bag (multiset): unordered, multiplicity

• set: unordered, uniqueness

• list (sequence): ordered, multiplicity

• unordered list: ordered, uniqueness