Does the Cartesian product of an infinite family have all the elements we expect?

Question

Given the axiom of choice, we know that the Cartesian product of an infinite family of non-empty sets is non-empty. However, this doesn't tell us whether the Cartesian product contains every element we might "expect".

For example, consider the Cartesian product of the family $\{A_i\}$ for $i \in \mathbb{N}$, where $A_i = \mathbb{N}$ for each $i$. Is it the case that every possible sequence of natural numbers appears in this Cartesian product?

More generally:

If we take the axiom of choice, can we make a general statement about the Cartesian product of an infinite family of non-empty sets that's stronger than the statement "it's non-empty"?

Answer 1

No. Generally, you can't say something stronger which isn't a consequence of the "weaker" formulation anyway.

Suppose that $\{A_i\mid i\in I\}$ is a family of sets. The product $\prod_{i\in I}A_i$ is a set defined using the separation schema. It is the set of all functions $f$ with domain $I$ such that $f(i)\in A_i$.

All of them. And the existence of this set has nothing to do with the axiom of choice. If $\{A_i\mid i\in I\}$ is a family whose product is empty, then its product is the empty set. Certainly if any set exists, it should be the empty set.

But all of them doesn't mean that any such function exists to begin with, so this set might be empty. The axiom of choice simply says, no it's not empty. If it isn't empty it has all the choice functions that the universe knows about.

To sum up, choice functions are like cockroaches. Where there is one, there are many. Usually infinitely many.

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There is a delicate issue here about different models of set theory which may know about more choice functions or less choice functions. But I think it might be counterproductive to start a discussion about that here.

Answer 2

Axiom of choice implies König's theorem which estimates cardinality of the product from below, namely $|\prod_{i\in I}A_i|\geq|\sum_{i\in I}A_i|$, the sum stands for the disjoint union. In particular, the product has at least as many elements as any of the factors, which is stronger than just non-empty.

However, if $I=\mathbb{N}$ and $A_i=A$ "every possible sequence" from $A$ is in the product by definition of the product regardless, even if the product is empty. When $A=\mathbb{N}$ there is no need even for the axiom of choice to claim non-emptiness since $\mathbb{N}$ is well ordered and we can pick $1$ from each copy.