The smallest uninteresting number is 198

A well-known joke/theorem is that all natural numbers are interesting. The proof goes as follows: assume that there exists a non-empty set of uninteresting natural numbers. Then this set has a smallest element. But that makes it interesting, so we have a contradiction. Incidentally, this proof applies to the integers and, with a bit of a stretch, to the rationals. It definitely does not applies to the reals, though, no matter how hard you believe in the axiom of choice.1

I was wondering, though, what is the smallest uninteresting number. It must exist, because we fallible humans are undeterred by the mathematical impossibility and simply do not find most natural numbers interesting.

Luckily, there is a objective criterion to determine whether a natural number is interesting: is there a Wikipedia article written about it? I then went through the Wikipedia articles about numbers, and found the first gap at 198. But now since this number became interesting, surely we should write a Wikipedia article about it?

This gives rise to another paradox: if we do write a Wikipedia article about 198 it will cease to be interesting, and of course we should delete the Wikipedia article about it. But this will make it interesting again, and we should again write the article.

You can see this paradox playing out in the revision history of the Wikipedia page: the article is indeed being repeatedly created and deleted.

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7 Responses to The smallest uninteresting number is 198

  1. Dimitris Papadimitriou says:

    Perhaps Jorge Luis Borges could have thought a clever short story based on this funny “theorem”.
    A more mundane way of confronting it is to “demand” a rigorous definition for ‘interesting’.

  2. Mateus Araújo says:

    I asked ChatGPT to do it, but the result was underwhelming. My approach would be to write about a monk cursed to go through each natural number in order finding number-theoretical properties that make it interesting. As the years go by each number becomes more of a struggle, and the monk is slowly descending into madness, until he hits 198. He tries in desperation to invent new number theory to make 198 be in any way remarkable, but all fails. He is at the point of taking his own life until he has the revelation that being so completely unremarkable from all possible points of view makes 198 actually quite interesting, it is the only number he has found with this property.

    The story ends with the monk’s face contorting in despair as he realizes: “the only number so far…”

  3. Dimitris Papadimitriou says:

    Yeah, something similar in spirit I had in mind, some characteristic Borges – style Legend, that could have been in one of his famous anthologies like “Ficciones” or “El Aleph”.

    It’s some kind of a relief, though, that chatGPT failed to adequately emulate his writings.

  4. Mateus Araújo says:

    Indeed. It would be very scary indeed if a newborn AI could already write on the level of one of the greatest writers we ever had.

  5. Abitbol says:

    The joke goes on like this:

    Theorem: All integers are uninteresting.
    Proof: Indeed, let us assume that there are interesting integers. Then there must be an integer which is the smallest among the interesting ones, which is therefore the smallest interesting numbers. Well, who cares?

  6. Ida Helena says:

    “Indeed. It would be very scary indeed if a newborn AI could already write on the level of one of the greatest writers we ever had.”
    Isn’t that the objective of A (Super) I developers? Or are you speaking about AI species…?

  7. Mateus Araújo says:

    Eventually, yes, that’s the goal. But right now GPT-4 is a newborn, and it cannot write on the level of Jorge Luis Borges, nor anybody expects it to.

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