I am wondering if statements which refer to generic numbers with indices have notions of order between these numbers. As an example of what precisely I am wondering, take this statement:$a_1 \cdot a_2\ldots \cdot a_{n-1} \cdot 2a_n = 2k $, where $k$ is an integer and $a_1, a_2, \ldots,a_{n-1}, a_{n} $ are natural numbers.
Is it correct to think of this as saying the product of any $n$ numbers where one of them is even is even, or the product of any $n$ numbers where the $n$th number is even is even? I am conflicted because when we have enumerated $a_1, a_2, \ldots,a_{n-1}, a_{n} $ , it seems we have presupposed an order to them by giving them indices, but on the other hand we usually think of numbers as forming sets and sets have no notion of order in them. So which of these 2 ways to think of this example are correct?
Read literally, your statement says that the product of any $n$ numbers, where the $n$th number is even, is even.
Since multiplication is commutative, you can immediately deduce from this that the product of any $n$ numbers, where one of them is even, is even.
If multiplication were replaced by some non-commutative operation here, this deduction would not be valid.