1

This question is motivated by Do we have $\mathbb{R} \times \mathbb{R}^{3}= \mathbb{R}^{4}$ or just $\mathbb{R} \times \mathbb{R}^{3} \simeq \mathbb{R}^{4}$. Other questions in this context are Is the Cartesian product of sets associative? and Associativity of Cartesian Product.

Of course there is a natural bijection $(X \times Y) \times Z \to X \times (Y \times Z)$ which imo allows to use the sloppy notation $(X \times Y) \times Z = X \times (Y \times Z)$. There are many explicit constructions of pairs and of Cartesian products, but I do not know any where we have $(X \times Y) \times Z = X \times (Y \times Z)$ in the literal sense.

Here is my question:

Does there exist a definition of $X \times Y$ making the Cartesian product strictly associative? Or can one prove that this is impossible?

Kritiker der Elche
  • 4,417
  • From the categorical perspective, the "set of ordered pairs" is just a construction of the concept $X\times Y$. $(X\times Y)\times Z\cong X\times(Y\times Z)$ is formally true, and equality can hold or fail to hold completely depending on how you encode, and construct, sets in your particular choice of foundations. I'm no set theorist but I believe this question is of the same vein as the question: "Is $\Bbb N$, as built by Von Neumann ordinals, equal to $\Bbb N\subset\Bbb Z$?" – FShrike Jun 09 '22 at 15:25

0 Answers0