# What is the substance/significance of the axiom "Sets are objects"?

Tao (Analysis I, 2016, p. 34) writes:

Axiom 3.1 (Sets are objects). If $$A$$ is a set, then $$A$$ is also an object. In particular, given two sets $$A$$ and $$B$$, it is meaningful to ask whether $$A$$ is also an element of $$B$$.

I don't quite understand what the substance/significance of this axiom is.

What is the point of declaring that sets are objects? Are there for example objects/things that are non-objects?

I think all he wants to say is that a set can be an element of another set. So why not just say that? What's the point of saying that "sets are objects"?

• That depends. Has he defined what an "object" is, in any way? Apr 14, 2020 at 8:30
• @Arthur: No (or I don't think so).
– user629687
Apr 14, 2020 at 8:33

The point here is that sets can be elements of other sets. In fact, they are always elements of other sets.

But, and this is a big one, first year students will not always recognise this immediately. Part of this is due to the fact that while the concept of a collection is perhaps as primitive to mathematics and abstract thinking as the concept of a number, we do not explain that to children.

So as a result of this, people come with a more "type theoretic" approach. Sets contains objects, they are not objects. Just like a number is "not a function". But mathematically speaking, sets are objects, and they are first-class citizens of the mathematical universe, just like any other object. And as such, it is their right to be elements of other sets.

Having taught first set theory courses for a few years I can say that a nontrivial percentage of students will have problems with this idea in the first two weeks. Of course, many others accept it intuitively and immediately. My point is, this is something that needs to be explicitly pointed out, and not left for the reader to conclude themselves. Especially when writing an introductory book on analysis.

In Tao's system there are different types of objects : sets, numbers, functions.

We have sets of numbers and sets of functions, and thus we can write e.g. $$n \in \mathbb N$$.

When Tao state the (IMO quite useless) Axiom 3.1: sets are objects, he want only stress the fact that also sets can be to the left of the "$$\in$$" relation.