One Original Thought: Proof by Diagonalization
Inspired by Anish's blog post: One Original Thought
"I probably don't have any thoughts that no one has thought about before."
The above statement is probably true, and people have probably had this thought before too.
And same with the above one.
In this paper, we prove the existence of an original thought via an explicit diagonalization argument. Unlike prior work, our method is reproducible.
Let \(T\) be the set of all thoughts beginning with:
"I probably don't have any thoughts that no one has thought about before me."
"Someone has likely had this above thought before."
Then succeeded by some infinite chain of either
\(A\): "And someone has probably had this above thought too."
or
\(B\): "Maybe no one has had this above thought."
For example:
"I probably don't have any thoughts that no one has thought about before me."
"Someone has likely had this above thought before."
"And someone has probably had this above thought too."
"Maybe no one has had this above thought."
"And someone has probably had this above thought too."
\(\ldots\)
is an element of \(T\). Observe that any element of \(T\) can be thought of as a function \(f : \mathbb{N} \to \{A, B\}\), where \(f(i)\) represents the \(i\)-th element of the thought. The example above begins \(f(1) = A\), \(f(2) = B\), \(f(3) = A\).
Let \(T' \subseteq T\) be the set of all such thoughts that have been had by a human before. We construct an original thought \(t_D : \mathbb{N} \to \{A, B\}\) in \(T \setminus T'\).
Observe that we can inject \(T'\) into \(\mathbb{N}\), where the map \(\phi(t)\) is the chronological order of when \(t\) has been thought. For example, if \(t\) was the first thought of the form \(T'\) ever thought, \(\phi(t) = 1\). Write \(t_i\) for the thought with \(\phi(t_i) = i\).
Define \(t_D\) such that
\[ t_D(i) = \begin{cases} B & \text{if } t_i \text{ exists and } t_i(i) = A, \\ A & \text{otherwise.} \end{cases} \]
That is, the \(i\)-th line of \(t_D\) flips the \(i\)-th line of the \(i\)-th thought ever had.
Visually,
| \(1\) | \(2\) | \(3\) | \(4\) | \(\cdots\) | |
|---|---|---|---|---|---|
| \(t_1\) | A | A | B | A | \(\cdots\) |
| \(t_2\) | B | B | A | A | \(\cdots\) |
| \(t_3\) | A | B | A | B | \(\cdots\) |
| \(t_4\) | B | A | B | B | \(\cdots\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) |
| \(t_D\) | B | A | B | A | \(\cdots\) |
We claim that \(t_D\) is original, i.e. is a thought not in \(T'\). Consider any \(t \in T'\). Then, \(t_D(\phi(t)) \neq t(\phi(t))\), meaning that \(t_D \neq t\).
Hence we have constructed an original thought. \(\blacksquare\)