Is there a way to construct larger cardinals without choice axiom?

Question

From Cantor's Theorem, we know that $|\mathcal{P}(X)| > |X|$. So, we can define inductively a set with cardinality $\aleph_n, \forall n \in \mathbb{N}$. Let $\lbrace A_i\rbrace_{i \in \mathbb{N}}$ be a family of sets with $|A_i| = \aleph_i$. If we take the cartesian product $\prod_{i = 0}^\infty A_i$, we get a set such that it's cardinality is greater than $\aleph_n, \forall n \in \mathbb{N}$. But this product is only well defined because of the choice axiom. So I'm wonderering if it's possible to construct a set with this property without the choice axiom.