# Honorable mention, or condensing responses

Months back I asked a question (Fields of Arbitrary Cardinality, sorry I'm on the phone app and can't immediately link it) on (i) whether it is true that any arbitrary infinite set $X$ can be endowed with a field structure, and (ii) whether choice-type methods were needed to demonstrate it. One responder provided such a construction involving an equivalence relation on the Kleene closure of $X$ (i.e. $X^* = \cup_{n \in \mathbb{N}} X^{n}$), where the equivalence classes under the relation of strings being identical up to permutation were at most finite. At the time, I gave this answer the check-mark, but we were uncertain whether choice had been used in the proof, seeing the statement that $X$ and $X^*/ \sim$ are equitotient as probably the only spot to be inspected, but unsure whether choice came into play.

Later, I read somewhere that the equitotience of $X^2$ and $X$ for all uncountable sets is equivalent to full choice, so the construction requires choice. Therefore we had a choice-dependent solution to (i). However, in the comments on this answer (where I "reported back" my findings on the choice-dependency of the particular construction), someone said that in fact the statement that every set admitted a group structure was equivalent to choice, so this answered (ii).

Therefore we had that the existence of field operations for every infinite set was equivalent to full choice, but the full commentary was distributed among several locations. The answer gave a construction but without certainty of its use of choice, my comment (already pretty far down in the comments) confirmed the construction's implicit use of choice, and another comment gave the reverse implication that fields (or groups) for every infinite set implied choice.

My concern is that as I understand MSE, questions are understood as having twofold purpose: (a) the OP gets his question answered, and (b) later users or non-users might be able to search MSE for answers to their question. The OP may use a check-mark to distinguish a solution. But in my case, anyone in situation (b) who found this question would have to do a good bit of digging to find the complete answer that was found. Is there a way the OP can better direct readers toward information contributed by commenters?

Thanks.

• Collect the tidbits into an answer. Make it community wiki if you want to make it easier for others to add further bits. Commented May 21, 2016 at 13:29
• As in provide my own answer?
– AJY
Commented May 21, 2016 at 13:32
• Yes. Or poke the commenter to write an answer, or the answerer to expand their answer. Commented May 21, 2016 at 13:36
• Thanks. Will do.
– AJY
Commented May 21, 2016 at 13:37