set versus collection

set versus collection

Based on the sources and our conversation history, the key distinction between a set and a collection arises from foundational issues in set theory, particularly when dealing with infinite collections.

Here’s a breakdown:

  • Informally, a set is often described as a collection of items. Another source defines a set as a collection of well-defined items.
  • For finite collections, the idea that any collection is a set seems reasonable. Examples of finite sets given include the days of the week, the factors of a number like 24, the set of natural numbers less than 10, and months in a year.
  • However, the sources explicitly state that not every collection is in fact a set. This becomes particularly problematic when dealing with infinite collections.
  • The reason for this caution stems from what is known as Russell’s Paradox. This paradox questioned the fundamental idea that any collection could be considered a set, especially in the context of trying to use set theory as a foundation for all of mathematics.
  • Bertrand Russell’s work demonstrated that the collection of “all sets” leads to a problem if one assumes it is itself a set. The paradox is illustrated by considering a collection, say ‘S’, defined using set comprehension as the subset of all sets that do not contain themselves. If we assume that the collection of all sets exists as a set, then this subset ‘S’ is also a set because it is defined by applying a condition (set comprehension) to an existing set (the set of all sets).
  • The paradox then asks whether this set ‘S’ belongs to itself.
    • If ‘S’ belongs to itself, then by its own definition (sets in S do not contain themselves), it cannot belong to itself. This is a contradiction.
    • If ‘S’ does not belong to itself, then by the condition used to define ‘S’ (sets in S are those that do not contain themselves), it must belong to itself. This is also a contradiction.
  • The conclusion from this paradox is that the initial assumption that leads to the contradiction must be false. The assumption that goes wrong is the assumption that there is a set of all sets. Therefore, not every collection can be called a set; specifically, the collection of all sets does not exist as a set.
  • In modern mathematics, when one is unsure if a collection satisfies the necessary properties to be a set, it is sometimes safer to refer to it as a class. A class is simply a collection of objects that may not have the implied properties expected from sets.
  • The sources advise that when working with sets, especially in this course, it is safe to assume that known collections of numbers like natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R) are sets. New sets should be constructed from these existing, well-defined sets using standard set operations like union, intersection, Cartesian product, and set comprehension. The caution is against “manufacturing sets out of thin air”, such as the problematic “set of all sets”.

In summary, while “set” is often used informally to mean “collection,” the sources highlight that in formal mathematics, particularly concerning infinite collections and foundational issues, not every collection is a valid set due to paradoxes like Russell’s Paradox. The collection of all sets is a key example of a collection that is not a set. The term “class” is sometimes used for collections that might not be sets.