![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png) simont |
Thu 2012-11-22 14:57 |
Indeed, there's no shortage of ways to show that two countably infinite sets are bijectable to one another. What strikes me as interesting (apart from the match of surnames) is that one reasonably natural way for a mathematician to do it (applying a theorem which happens to be more general than needed) also gives rise to a method of quoting which is not merely plausible for practical use but actually in use.
Of course, there's no need in general for the transformation from a string to its quoted form to be bijective, or even to be a function (by which I mean 'single-valued'). There are plenty of practical reasons why you might want to have some strings not be legal in the quoting syntax even though they aren't among the set of strings you really needed to avoid, or why you might want to provide multiple equivalent ways to quote the same thing (e.g. \042 vs \x22 vs \" in C string literals). Bijectivity is desirable only because it correlates with space-efficiency, and only as long as it doesn't introduce any other inconveniences. For instance, the qmail-style mbox quoting transform could have been replaced by simply prefixing every non-separator line with a > sign, but that wastes more space for a reason not totally unconnected to its non-bijectivity.
So the Gödel-numbering approach probably would be too inconvenient to be worth the space saving, if it didn't turn out to reduce to anything more easily stated! |
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