In accordance with our informal image of the hierarchical way sets are constructed.
there is no "sets of all sets" i.e., there is no sets having all sets as members.
In the future sections, we will prove this statement; that is, the nonexistence of a set of all sets will become a theorem, provable from the axioms.
| a German logician and mathematician | a German-born Israeli mathematician . |
| known for his proof of the well-ordering theorem. | best known for his work on axiomatic set theory |
The collection cannot be a set, but what status can we give it? Basically there are two alternatives:
The Zermelo-Fraenkel alternative: the collection of all sets doesn't need to have ontological status at all, and we never speak of it. By ontological we mean that we just have ignore that "collection of all sets" exists. Thus in Zermelo-Fraenkel set theory "collection of all sets" does not exist. But at times that we have to speak about it, we seek rephrasing just to avoid it.
The von Neumann-Bernays alternative: collection of all sets can be called a class. Likewise, other collection of sets are called a class; that is, any set is a class, but some classes are too large to be sets.
The Zermelo-Fraenkel alternative seems to be the better of the two in advanced works in set theory. It profits from the simplicity of having to deal with only one sort of objects (set) instead of two (classes and set).
Disclaimer: this is a summary of section 1.3 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshot from generated latex form using typora.
Additional readings:
[1] Alternative Axiomatic Set Theories
[2] Von Neumann–Bernays–Gödel set theory
[3] Zermelo–Fraenkel set theory
Thank you for reading ...
