In a script I found a recursive defintion for the cartesian product of n sets:
With $P_n = \{i ∈ N_+ ∣ i ≤ n\}$
the cartesian product of sets $M_i$ with $i \in P_n$ is defined as follows:
$\times_{i\in P_1} M_i = M_1$
$\times_{i\in P_{n+1}} M_i = (\times_{i\in P_n} M_i) \times M_{n+1}$
Note: It should be a large "cross" for the cartesian product, but \bigtimes does not work in this environment.
Is this definition correct? I don't understand the subscript $i\in P_{n+1}$. For $n=4$ you could chose any $1\leq i\leq 4$, but how does the definition work out if I can arbitrarily chose this value?
the definition is correct.
$\times_{i\in P_1} M_i = M_1$
$\times_{i\in P_{n+1}} M_i = (\times_{i\in P_n} M_i) \times M_{n+1}$
then for your case when $n=4$ we have,
$ \begin{split}\times_{i\in P_{5+1}} M_i &= (\times_{i\in P_5} M_i) \times M_{6}\\ &=(\times_{i\in P_4} M_i)\times M_{5} \times M_{6}\\ &.\\&.\\&=(\times_{i\in P_1} M_i)\times M_{2}\times M_{3}\times M_{4} \times M_{5}\times M_{6}\\ &=M_{1}\times M_{2}\times M_{3}\times M_{4} \times M_{5}\times M_{6}\end{split}$