# What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.

My question is what the definition of a set is?

I have noticed that many other definitions start with a set and then something. A group is a set with an operation, an equivalence relation is a set, a function can be considered a set, even the natural numbers can be defined as sets of other sets containing the empty set.

I understand that there is a whole area of mathematics (and philosophy?) that deals with set theory. I have looked at a book about this and I understand next to nothing.

From what little I can get, it seems a sets are “anything” that satisfies the axioms of set theory. It isn’t enough to just say that a set is any collection of elements because of various paradoxes. So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?

Formally speaking, sets are atomic in mathematics.1 They have no definition. They are just “basic objects”. You can try and define a set as an object in the universe of a theory designated as “set theory”. This reduces the definition as to what we call “set theory”, and this is not really a mathematical definition anymore.

In naive settings, we say that sets are mathematical objects which are collections of mathematical objects, and that there is no meaning to order and repetition of the objects in the collection.

And when we move back to formal settings, like $$ZF,NBG,ETCS,NF\sf ZF,NBG,ETCS,NF$$2 or other set theories, we try to formalize the properties we expect from sets to have. These may include, for example, the existence of power sets, or various comprehension schemata. But none of them is particularly canonical to the meaning of “set”.

These are just ways to formalize, using a binary relation (or whatever you have in the language), the idea of membership, or inclusion, or whatever you think should be the atomic relation defining sets. But as for a right definition? In this aspect “set” is as Platonic as “chair” or “number” or “life”.

Footnotes:

1. This assumes that you take a foundational approach based on set theory. There are other approaches to mathematics, e.g. type theory, in which the notion of “type” is primitive, and sets are just a certain type of objects.

Sufficiently strong set theories can interpret these foundations as well, reducing them to sets if you choose to, or not if you choose not to.

2. These are Zermelo-Fraenkel, von Neumann-Goedel-Bernays, Elementary Theory of Category of Sets, and New Foundations. These are not the only set theories, of course. And the point of the answer is that these just offer formal frameworks for the notion of “set” as a primitive object (in one way or another).