# Why can't union be recursively defined (Tao's Analysys I)

I'm currently working on Terence Tao's Analysis I, and in page 42, Tao defines pairwise union:

Axiom 3.4. Guven any two sets $$A,B,$$ there exists a set $$A\cup B,$$ called the union $$A\cup B$$ of $$A$$ and $$B,$$ whose elements consists of all the elements which belong to $$A$$ or $$B$$ or both.

However, in page 43, he gives an idea to form triplet, quadruplet, etc sets by doing $$\{a,b,c\}:=\{a\}\cup\{b\}\cup\{c\}$$ and so forth (he begins this Chapter by stating the pair sets axiom), yet he claims that

we are not yet in a position to define sets consisting of $$n$$ objects for any given natural number $$n;$$ this would require iterating the above construction "$$n$$ times", but the concept of $$n$$-fold iteration has not yet been rigorously defined.

I'm not sure what he means by that...I looked for that concept and found that it means functions that take other functions as inputs OR have as a result another function...but I don't see the relation...

• Has he proven yet the existence of $\mathbb N$ with it's fundamental property "every non empty subset has a minimum"? If not you cannot argue by induction! Apr 20 at 18:40
The fact that you can define triplets, quadruplets, etc. is an informal observation, but you don't yet have the tools necessary to formalize the general concept of an $$n$$-element set. After all, what is $$n$$? What is a natural number? And what does it mean to repeat or iterate an operation? (What is an operation?)