Notation for element of an ordered tuple?

Question

When $X$ is a set, we can write:

for all $x\in X$ ...

But say that $X=(a, b, c, ... ,n)$. I.e. an ordered tuple.

Is it standard notation to still say the following?

for all $x\in X$...

It might be confusing because if you interpret it as a sentence in ZFC, then you're not quantifying over the thing you want (you're including the sets representing the order). But how would we write down to quantify over just the elements of the tuple?

Answer 1

I've seen it written as you wrote it, numerous times. Sometimes conciseness yields better clarity than perfect rigour.

An alternative is to write the elements of the $t-$uple in an "indexed" form such as $a_1,\dots,a_t$, and then quantify over the range of indices ("$\forall i$ ... $a_i$ ...").

Answer 2

If we insist on rigor here, we can proceed in a slightly ugly way.

If we define a tuple as a function from sets of integers into the target set then we can proceed as follows.

Let $$ [n] = \{m \in \mathbb{N}: m<n\} = \{0, \ldots, n-1\} = n $$ Where the last equality follows if the natural numbers are defined as the Von Neumann ordinals.

We define an $n$-tuple $X$ over the set $S$ as the function

$$ X:[n] \to S $$

Then the $i^{\text{th}}$ element of $X$ where $i \in [n]$ can be found by $$ X(i) = X_i $$

We write $X= (X_0, \ldots, X_{n-1})$ with $X_i \in S$. If we want the set $\{X_0, \ldots, X_{n-1}\}$ then we can take the image of $[n]$ under $X$. That is

$$ X([n]) $$

So we could write, for example,

$$ \forall s \in X([n])\ldots $$

If you want you could introduce a special notation for this like $$ \text{Set}_n(X), \text{Set}(X), \text{Img}_X([n]), \text{Img}(X) $$ to indicate where converting the tuple $X$ into the set containing elements of its range.

None of this is really standard or pretty but I guess it's possible. I answer here because I had the same question as you and this is as far as I've gotten.