I was scanning the blogosphere late one night last week, amazed at the humongous number of blogs, each blog listing other blogs on their blogrolls; and upon searching items in the blogroll, one is led to still other blogs, with yet more blogs listed on blogrolls* with little repetition. It is pretty overwhelming. Do people manage to read this stuff? I mean, the current blog is a niche audience, but most of the others I saw pretty much review the same current events (with slightly different spins): Romney, Superbowl, Facebook IPO, Planned Parenthood, too much snow in UK, Shieldcroc. One blog included on its blogroll list a blog named “There’s At least One Blog That Nobody Reads,” which I obviously didn’t read.
Studying the blogrolls I noticed that occasionally, perhaps in error, a blog would actually list itself in its own blogroll. Those self-referring blogs were rare though (not that my search was exhaustive, though it was exhausting.)
Always on the look-out for just such a phenomenon (if only to show students it can really happen), I immediately got the idea to start a blog, call it Goedel’s Blog (G), that would list, on its blogroll, all blogs that do not list themselves on their blogrolls (say as of a fixed date). Any blog that does mention itself on its
blogroll would not be included in blog G’s blogroll. So Blog G’s blogroll lists all and only names of blogs that do not refer to themselves in their own blogrolls.
Now you know the problem that arises:
Should Blog G list itself on its blogroll? If we list G on G’s blogroll, then we must take it off the list; whereas if we leave G off G’s blogroll, then we must put it in.
If your job is to create blog G, including G’s blogroll, what do you do?
*Reminder: Sailor, maybe start an official blogroll in this blog.
Yes, the following is a logical contradiction:
G is included in G’s blogroll if and only if G is not included in G’s blogroll.
G iff ~G
This may be the most perfect example I have found of this (although I’ve collected several others over the years that aren’t as realistic). Can it be avoided in the manner of Russell’s and similar paradoxes (of self reference)? I invite logicians to weigh in on this.
Submitted on 2012/02/07 at 11:41 pm
The task is a logical contradiction; can’t be done!
Ah, right: ≥ and ≤ can’t be used directly.
Submitted on 2012/02/07 at 3:20 pm
The analogy between “a blog whose blogroll lists all blogs that do not list themselves on their blogrolls” and Russell’s paradox, “the set of all sets that do not contain themselves” (and applied versions), is: set blog; set inclusion listing on a blogroll.
In the second, I’m alluding to Goedel’s second incompleteness theorem. The analogy is: consistent formal effectively generated theory blog; formula post; broken links absence of proof.
Submitted on 2012/02/06 at 2:39 pm
Not sure about your second point; I don’t see where Goedel does any such analogous thing, but I may be missing your idea. On your first, you’d need to show how this commits the violations that Russell’s paradoxes do. How would you do this?
Submitted on 2012/02/06 at 12:23 pm
Surely that should be Bertrand Russell‘s blog, not Goedel’s blog?
On Goedel’s blog there would be a post that purports to prove the consistency of Goedel’s other posts, but any link to that post from within Goedel’s blog would be broken.