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<p>This might be a good jumping-off point to talk about what it means for something to make sense intuitively, as part of a larger discussion about what it means for someone to be “good at math”. For example, I might have posed my questions somewhat differently, in which case more people might have sooner figured out what I was trying to get them to think about.</p>
<p>Which set is larger, the even integers or the integers?</p>
<p>If someone has been given a simple definition of infinity, some of what’s been discussed here might not make sense intuitively. OTOH, if they’ve been encouraged to explore the relations between sets, it might.</p>
<p>At any rate, my goal here is to provide a framework for discussion about acquiring mathematical skills. When talking about people who do mathematics for a living, not only do they answer difficult questions, they also ask them. They take a problem that is known and modify it in other ways, e.g. after determining the size of a set they ask questions about subsets of that set, and so forth.</p>
<p>FWIW, there are plenty of web sites that give answers to these and related questions, if you do a little googling or hunting around wikipedia.</p>