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 Phil-Infinity):
Course Description. PHIL-∞ is a (very high-level) philosophy research seminar. Students participate in term-paper 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, weak-gravitational 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 term-paper assignment:
- Write down the title of each course in the philosophy department. For example, if PHIL-101 “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 term-paper 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 hard-working 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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