The Web Page That Cannot Exist

I discovered by an accidental click that the web page for the 2019 annual meeting of the Linguistic Society of America (January 3–6, in New York) contains a hyperlink to itself (it’s in the paragraph about the Pop-Up Mentoring scheme). This almost certainly happened through a copy-and-paste error. It is not good web practice: Clicking on a self-link simply takes you to right where you are already, so it only wastes your time. But it reminded me of a wonderful illustration of a deep logical paradox devised by my friend Brian Rabern, a philosopher at the University of Edinburgh.

It would be a good thing, Rabern mused one day, if someone supplied some guidance about this web-design issue. And it would be easy to write a web-searching program to find all the pages that are free of links to themselves: For a page with the URL https://xxx.yyy.zzz, you would simply examine its HTML source to make sure that the string <a href="https://xxx.yyy.zzz"> did not appear anywhere in its HTML code. (To be fully general it should also be checked for <a href="">, because a link giving no destination URL also functions as a self-link.) Any page that meets the test of not containing a self-link could be cited as an example of good practice. A list of such pages could be automatically compiled. We could publish it on the web: a page giving links to every page on the web that is free of self-links.

There is nothing impractical about the programming for building this page. And the web is finite; the list would be large, but not infinite in size. Google’s index of the web is far larger, and is searched thousands of times every second.

So the task is feasible. Except for one cosmic-size, disastrous problem: It is logically impossible for the page to exist.

There can never be a web page listing every self-link-free page, for a very simple reason. Let’s imagine that The Chronicle of Higher Education was willing to host the page, and the plan was to use https://www.chronicle.com/no_self_links.html as its URL. To complete the page, we would have to decide whether to put a link to https://www.chronicle.com/no_self_links.html onto the end of our list or not. And there we would confront a true logical crisis. You see why?

If we added a line giving that link, then the page would have a link to itself, which means it should be ineligible to be on the list.

But if we leave it off because it is ineligible, then it is a page without a link to itself, hence an example of the good practice we want to recognize, which means it is eligible, and we would be duty bound to add it. Though of course that would mean we had to take it off again.

In other words, if the link is on the list then it mustn’t be, but if it isn’t then it should be. Contradiction.

Rabern’s paradox is deep and profound. Philosophers will see (as of course Rabern does) that it is basically a concrete modern variant of Russell’s paradox. In June 1901, Bertrand Russell discovered a contradiction lurking deep within the assumptions of the work on the foundations of mathematics on which Gottlob Frege had been laboring for years. Frege had been assuming that for every property there is a collection of things that have that property, and he needed that assumption. But it is not true, Russell realized. Take the property of being a collection that is not one of its own members. The collection of all collections that are not members of themselves cannot exist.

In 1902 Russell wrote to point this out to Frege. His respectful note is published in Jean van Heijenoort’s superb anthology From Frege to Gödel (Harvard University Press, 1967), and so is Frege’s reply (which was amazingly courteous and not at all defensive; all academics could learn from Frege’s willingness to accept that a serious error had been found in his work).

What’s particularly nice about Rabern’s paradox, for teaching purposes, is that it is not couched in abstract terms of set theory (e.g. collections that can contain collections as members). It is so much more concrete. Rabern regularly has to teach arcane aspects of logic to philosophy undergraduates who are taking it as a required course. It is far easier to get them to see the relevance of this weirdly uncompletable list of web links than to reflect on something as imaginary as the set of all non-self-membered sets. And sometimes, when teaching subjects of the kind Rabern and I teach (we are in a School of Philosophy, Psychology and Language Sciences) the heuristic power of the example chosen can be all-important.