Yes, I know of the value of zero (and not just because I studied theory/math heavy fields in uni).
Does not change that this does not always apply. Having no piece of paper is valid as "no PhD", but an empty piece of paper is not a valid PhD. In the same sense, an empty piece of paper is also not a "$0 banknote", it's not a banknote, period.
It is true that in many cases, empty files are valid for their context (e.g. CSV, plain text files, shell scripts), in many other cases, they are not (e.g. any file format at all that requires a magic number in a header or footer, which is many many file formats). Try to open these in those contexts, and you would usually get an error, for example: "You told me this was supposed to be a JPEG, but this file does not start with ff d8 ff e0 [Trivially true for an empty file! No, files are not equivalent to sets in the Zermelo-Fraenkel sense!] What is wrong with you?"
I don't get the discussion here. This is all obvious, and it was obvious in the original sentence from context.
I mean, if we want to run with the man analogies...
The empty set is a valid set. Perfectly reasonable, and completely necessary in many cases.
But the empty Group is not a valid Group. A Group requires an identity element, which means non-empty. The smallest valid Group, the Trivial Group, still requires an element.
I would argue a diploma or a banknote resemble a Group more than a Set. There are required elements, such as an issuing institution.
Does not change that this does not always apply. Having no piece of paper is valid as "no PhD", but an empty piece of paper is not a valid PhD. In the same sense, an empty piece of paper is also not a "$0 banknote", it's not a banknote, period.
It is true that in many cases, empty files are valid for their context (e.g. CSV, plain text files, shell scripts), in many other cases, they are not (e.g. any file format at all that requires a magic number in a header or footer, which is many many file formats). Try to open these in those contexts, and you would usually get an error, for example: "You told me this was supposed to be a JPEG, but this file does not start with ff d8 ff e0 [Trivially true for an empty file! No, files are not equivalent to sets in the Zermelo-Fraenkel sense!] What is wrong with you?"
I don't get the discussion here. This is all obvious, and it was obvious in the original sentence from context.