The element relation

Question

I am studying set theory and I have the following question: Are all the mathematical objects which we can quantify over, in essence, sets? I ask you this also because I wish to know if, when we say that an object is an element of a set, that object is also set. Thank you in advance.

Answer 1

As Will already said, in set theory every object we consider is a set.

• Every element of every set is a set,
• every natural number (in a precise way) is a set, e.g. $0 = \emptyset, 1 = \{ \emptyset \}, 2 = \{ \emptyset, \{ \emptyset \} \}$,
• every field, every vector space, every function, every topological space ... is a set,
• even every formula is a set.

The remarkable thing about set theory (though not at all a coincidence, it's been designed with precisely this intention) is that this theory is able to capture a great deal of mathematical concepts in a way that once all is said and done set theorists and mathematicians who don't think about a given mathematical concept as a set agree about its properties/theorems.