Self-descriptive lists—a short investigation

Extract

The way this problem is stated tempts one to think of it as an isolated puzzle. But there are a number of reasons why the problem may be worth investigating more closely. Perhaps you found it so mind-boggling that you couldn’t do it at all. Or you may have succeeded in finding one such list, but could not find a convincing proof that your list was the only possible solution. In either case it would have been worth having a go at a simpler version of the same problem first. But even if you solved the puzzle completely, the mathematician in you should be mildly inquisitive as to why it asks for a list of eight whole numbers, rather than say seven, or nine, or even six hundred and thirty four.