Let me quickly explain what I mean with my question.

Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in I}\kappa_i$ as follows: take collection $(A_i)_{i\in I}$ of sets such that $|A_i|=\kappa_i$, and let $A=\{(a,i):i\in I,a\in A_i\}$. Then we say $\sum\limits_{i\in I}\kappa_i=|A|$. We can do similar trick with multiplication, by setting $B=\{f:f\text{ is a function from }I\text{ such that }f(i)\in A_i\}$.

If we assume AC, then these notions are well-defined: First of all, we *can* choose such a sequence $(A_i)_{i\in I}$, and if we take two collections $(A_i)_{i\in I}$, $(A_i')_{i\in I}$ such that $|A_i|=|A_i'|=\kappa_i$, then we can use choice to get bijections between $A_i$ and $A_i'$ for every $i\in I$ and then use there to create bijection between $A=\{(a,i):i\in I,a\in A_i\}$ and $A'=\{(a',i):i\in I,a'\in A_i'\}$, so this procedure defines only one cardinal. Similar trick for multiplication.

However, if we do *not* assume choice, then 1. we don't know a priori that we can always choose any sequence $(A_i)_{i\in I}$, and even if we can, we have no guarantee that there is only one possible size of $A$ we can get in this way. I don't know of scenario where we cannot find any sequence of sets with given cardinalities, but I know that uniqueness of size can fail quite spectacularly, even if we add $2$ countably many times (see here).

I have three questions:

- Is it possible in abscence of choice that there is a collection of cardinals $(\kappa_i)_{i\in I}$, but there is no collection of sets $(A_i)_{i\in I}$ having respective cardinalities? (I suspect the answer is "yes")
- If the answer to 1. is yes, then is it known that existence of such a collection implies axiom of choice? (I suspect answer to this to be "doesn't imply" or "not known")
- Does existence of collections like above and "value of the sum and product of cardinals doesn't depend on choice of collection" imply axiom of choice? (no clue, but I'm hoping for "yes")

These are main questions I'm interested in, but we can ask more by asking which combinations of below three imply choice:

For any collection of cardinals, there is collection of sets of respective cardinalities. Sum of cardinals, if defined, is well-defined. (i.e. sum doesn't depend on choice of collection, but we allow possibility that the collection doesn't exist) Product of cardinals, if defined, is well-defined.

Thanks in advance for all feedback.