I wish to note that in the comment that prompted this question, I was paraphrasing Kaplansky from memory, not quoting him. Since there is now quite a discussion, people may be interested in the exact quote. Here it is, from Set Theory and Metric Spaces, p.21:
[The context is that he has just given the usual "naive" argument that any infinite set has a countably infinite subset.]
"(I can hear many readers snorting that, without any warning, I have sneaked in the countable axiom of choice. My reply is: guilty as charged. In an account of set theory designed for an apprentice mathematician, I think it is out of place to fuss with the countable axiom of choice. Historical evidence is on my side. It was only when the scrutiny of uncountable sets began that the axiom of choice got placed on the agenda.)"