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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...

Please advise, thanks a lot!

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    $\begingroup$ 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! $\endgroup$ Commented Apr 20, 2023 at 18:40
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    $\begingroup$ It sounds like you'll enjoy some of the upcoming axioms: Axiom 3.7 which gives the natural numbers; Axiom 3.11 which defines arbitrary unions. Keep in mind, as you read, that Tao is mixing formal development with informal intuition in order to aid understanding. Perhaps it would be very good to keep track, as you read, of what has already been formally defined or axiomatized, and what has not yet been. You can look for and find lots of things outside of the book, but that won't help you learn how to put those things in logical order. $\endgroup$
    – Lee Mosher
    Commented Apr 20, 2023 at 20:27
  • $\begingroup$ I was actuallly pretty confused, because in page 26 via Proposition 2.1.16., he "proves" that we can define things recursively. I should've known this was a quite informal approach and of course independent of this chapter. $\endgroup$
    – Vin300522
    Commented Apr 21, 2023 at 0:41

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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?)

There's a lot of logical machinery like this that we use freely in an informal way but which takes much more work to formalize. Formalizing means establishing definitions in the language of set theory that model the logical properties of the desired objects (using the logical properties of sets, which come from the axioms of set theory), so that we can study them as mathematical objects and formally prove things about them.

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  • $\begingroup$ I was actuallly pretty confused, because in page 26 via Proposition 2.1.16., he "proves" that we can define things recursively. I should've known this was a quite informal approach and of course independent of this chapter. $\endgroup$
    – Vin300522
    Commented Apr 21, 2023 at 0:41

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