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It doesn’t really matter, but I’m not convinced by the “common theme” sentence… . Surely the point is more like that it describes something which cannot be counted?
Mostly it means something larger than every natural number, although not always. I put a sentence saying that and noting a couple of exceptions (one each way). You could still argue (even with those exceptions) that it means something larger than whatever can be counted, for some sense of counting.
“Larger than” or “beyond”. I don’t think of ordinary real numbers as things one counts, but when thinking about calculus, I do think of and as ideal points beyond all finite real numbers. Similarly with valuation fields, etc.
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