PHIL∞: An Infinite Seminar?
Can every philosophy seminar be correctly completed in either finite time or infinite time? Here’s a (gödelesque) answer in the negative, in the form of a philosophy seminar I’m calling PHIL∞ (pronounced PhilInfinity):
Course Description. PHIL∞ is a (very highlevel) philosophy research seminar. Students participate in termpaper research, with the goal of determining the truth or falsity of the following claim:
All courses in the philosophy department can be completed in finite time.
(Technical note: students can neglect relativistic corrections to the notion of absolute time, since all coursework is guaranteed to take place in roughly the same inertial, weakgravitational regime.)
Course Requirements. In order to complete the course, students must correctly complete all the required coursework. The only required coursework in PHIL∞ is the following termpaper assignment:
 Write down the title of each course in the philosophy department. For example, if PHIL101 “Intro to Philosophy” is listed as a philosophy department course, then students must write this down. Obviously, PHIL∞ will also have to appear somewhere on their list.

For each course on the list formulated in 1, do one of the following:
 If the course requires infinite time to complete, then write down ‘infinite’ below the course title.
 Otherwise (if the course can be completed in finite time), then write down ‘finite’ below the course title. Then complete the termpaper assignment for that course by writing it out completely.
Remark. By enrolling in this course, students are making a very unfortunate mistake indeed. If you haven’t taken a moment to imagine what it would take to complete PHIL∞, try it now. (I go through it below.)
Imagine a hardworking student, who writes down each course on the department listing, and below each one either writes ‘infinite,’ or else writes ‘finite’ and does the term paper. But what does one write below PHIL∞? It would be incorrect to write down ‘infinite,’ because then the student would have completed the course in finite time.
Therefore, the student must write down ‘finite,’ and then rewrite the term paper requirement for PHIL∞. This entails writing down all the department courses, and so on.
But sooner or later, the student will get to PHIL∞ again, and start the cycle all over. This cycle would continue indefnitely, so it would also be incorrect to write down ‘finite’ below PHIL∞.
So it appears that there is no correct way to complete PHIL∞, in finite or in infinite time?.
The question I leave to you then, is: why should that be the case?
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This hardworking student will finish the course in finite time. Here’s how:
Suppose you have written out the term paper requirement for every course other than phil infinity and correctly labeled all other courses as “infinite”. Below Philinfinity, write, “finite”. Assume that all of this is written on a single sheet of paper (you may either imagine a long scroll of paper or microscopic font). Roll the paper by connecting the side with the head of the document to the side with the final “finite”. Use tape or some sort of glue. Submit the rolledup paper. Graduate with honors.
Fantastic. Jonathan would clearly pass my infinite class.
My idea was simply to have a class that iterates indefinitely iff it’s classified as finite, and terminates otherwise. Jonathan got around this first horn by turning in an assignment with a nontrivial topology. Clearly, my course description requires a more precise description, in order to insulate against tricky students like this.
Thanks!