In the accepted response in this thread, Asaf Karagila wrote that
Note that it is perfectly possible to choose from infinitely many sets without the axiom of choice under a severe constraint that they have some common characteristic. From infinitely many sets of natural numbers we can choose the minimal in each set; from infinitely many finite sets of real numbers we can choose the maximal element of each set; etc etc.
For example, in the original post in this thread, Framate took the set $Y$ (which was not yet known to be finite) and selected from the preimage of every element $y\in Y$ the element with the least index using the well-ordering principle.
I have two questions:
- Regarding the "severe constraints" that Asaf Karagila wrote about, can we say anything general about what such constraints allow for bypassing the axiom of choice?
- One such severe constraint is that all of the sets are totally ordered and each has a least element in that total order. If there was just one such set, then it makes sense that we can choose its least element. But what rule exactly allows us to make this choice when there are more sets, say infinitely many? It seems like doing this requires some power, but I don't know the source of that power.
Best Answer
Your first question is poorly defined IMO, so I'm not sure how to answer it, but perhaps the answer to your second question will shed some light on the first.
Suppose $\{S_i\}_{i \in I}$ is an indexed family of subsets of $\mathbb{N}$. We can construct the mapping $m: I \to \mathbb{N}$ giving for each $i$ the minimal element of $S_i$ by constructing the set $m := \{(i,n) \in I \times \mathbb{N} \mid n \text{ is the minimal element of } S_i\}$. This perfectly doable without choice ans $m$ is a function under the set theoretic definition, since for each $i$, there will be a unique $n$ s.t. $(i,n) \in m$.