> So presumably any definition should be compatible with our existing concepts. Yes, this is the wrong way (IMO) to think about it. The right way is to notice

> So presumably any definition should be compatible with our existing concepts. Yes, this is the wrong way (IMO) to think about it. The right way is to notice that it is a definition, and to think about it like every other definition. It has the name "same size", because it has some properties common with "same size" of finite cardinalities, but not all properties common. And this is the proper way to think about every other definitions and objects as well, even if they are technical terms, or borrowed from common English. (BTW I believe that rings are called rings, because of some similarities with rings.)