How best to mathematically describe 'Tristram Shandy problem'

Question

This question is related to a question posted on philosophy.SE, but I am posting it here since it is mathematical in nature, and since answers using LaTex are available here. (I will be providing an answer myself for the benefit of philosophy.SE users, but I also invite other answers.) The question concerns cases where a task, and the time available for that task, can be represented as cardinally equivalent sets, yet it is impossible that the task could ever be completed:

Well known is the story of Tristram Shandy who undertakes to write his biography, in fact so pedantically, that the description of each day takes him a full year. Of course he will never get ready if continuing that way. But if he would live infinitely long then his biography would get 'ready', because every day in his life, how late ever, finally would get its description. No part of his biography would remain unwritten, for to each day of his life a year devoted to that day's description would correspond." [A. Fraenkel: "Einleitung in die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24. A.A. Fraenkel, A. Levy: "Abstract set theory", North Holland, Amsterdam (1976) p. 30]

It is clear to me that Tristram Shandy can never finish his bioigraphy, even if he lives infinitely. With this in mind, how is this problem best described mathematically, in a way that allows the different concepts in the problem to be adequately defined (e.g., finishing the biography, finishing a particular day, etc.) Once described, how do the different concepts relate.

Answer 1

The problem can be described as follows. Let $n \in \mathbb{N}$ be the number of days that Shandy has lived and let $D(n)$ be the number of days that are written, so that $U(n) = n - D(n)$ is the number of days that are unwritten. From the specification of the problem we have the following properties:

$$\begin{matrix} \begin{aligned} &\text{Every day eventually gets written} & &(\forall n \in \mathbb{N}): D(n) < \infty, \\[6pt] &\text{He never finishes writing his biography } & &(\forall n \in \mathbb{N}): D(n)<n, \text{ } \\[6pt] &\text{The written part increases without bound } \quad \quad \text{ } \text{ } & & (\forall u \in \mathbb{N}) (\exists n \in \mathbb{N}): D(n) \geqslant u, \\[6pt] &\text{The unwritten part increases without bound } \quad \quad \text{ } \text{ } & & (\forall u \in \mathbb{N}) (\exists n \in \mathbb{N}): n-D(n) \geqslant u. \end{aligned} \end{matrix}$$

There is no inconsistency between saying that Shandy will never finish his biography, and saying that any given part of the biography will eventually be written. Intuitively this occurs because the written component increases without bound, but the unwritten component also increases without bound. Does anyone have a better way to describe this?

Answer 2

For simplicity, I assume every year has exactly 365 days.

The main mathematical content here is the definition of a 365-1 correspondence between the natural numbers and itself:

• $W(m, n)$ is the relation "On day $m$, Tristram is writing what happened during the $n$-th day of his life"

which could be expressed more algebraically as

• $W(m,n) :\equiv (n = \lfloor \frac{m}{365} \rfloor) $

That such a thing can exist is a striking feature of infinite sets. This fact (or something closely related) is presumably the intended message of the parable.

The form of the story is to mimic a way you would prove such a correspondence can't exist for finite sets, so that it can highlight how a specific form of reasoning you'd use in the finite case goes wrong, and how to arrive at the correct answer instead.

If we use a similar construction but change it so that Tristram Shandy only lives for, say, 7300 days, he only has time to write down 20 days of his life during this interval. There have to be 7280 days unaccounted for! So the relation $W_{7300}$ we construct cannot total in its second argument; there are 7280 different values of $a$ less than 7300 for which $W_{7300}(m,a)$ is never true.

The story describes this as "the autobiography is unfinished", because there unwritten days — values of $a$ for which $W_{7300}(m,a)$ is never true.

We might imagine that one could make sense of an argument for the general case by "taking the limit", but the fact we have an explicit 365-1 correspondence between the natural numbers and itself shows that such a limiting argument doesn't work.

In particular, for every $n$, there exist $m$ for which $W(m,n)$ is true. Since every $n$ is accounted for, the story describes this as "the autobiography is finished".

The mistake is focusing on the question "On day $m$, how many days remain unwritten?" and the fix is to go back to what is actually being asked: "Are there any days on which day $n$ gets written?"