Math puzzle: minimum number?


What is the minimum number that cannot be expressed with less than two english words? Also, how about less than thirteen english words?

Comments (12)

  1. Jonathan says:

    "Minus infinity"?

  2. David Betz says:

    I was thinking minus infinity too…

  3. kfarmer says:

    According to my number theory class — ages ago — infinity is NaN. It’s more of an innumerable.

  4. AdiOltean says:

    Good answer on minus infinity. 🙂

    Now, how about the thirteen words problem?

  5. Matthew says:

    Minus ninety

  6. Mr Blobby says:

    "The least number expressible in under thirteen words" is a valid English phrase using only 8 words. Therefore if there were ever such a number this would be a contradiction.

    This is a special case of Russell’s paradox, and is resolvable using his theory of types.

  7. Mr Blobby says:

    Sorry, "expressible" should obviously say "inexpressible" up there. We can make it watertight by changing the phrase to "The least integer number inexpressible using less than 13 English words".

    This rules out the case of infinities as with the examples above.

  8. AdiOltean says:

    Allright – Mr. Blobby found the correct answer for the second question. The statement "The least integer number inexpressible using less than 13 English words" does not makes sense as a definition of a given number, since it is self-contradictory. It is like saying "the minimum number that is smaller than 1 and bigger than 2".

    As for the first question: minus googolplex?

  9. Mr Blobby says:

    A interesting puzzle based on this:

    "The least integer indescribable using less than ten words" leads to the same contradiction as with thirteen words.

    But "The least integer indescribable using less than two words" does not, because there is no phrase with the same meaning under two words long.

    So the question is this: what is the maximum n such that "the least integer indescribable using less than n words" does not lead to a contradiction?

    [I don’t know the answer!]

  10. AdiOltean says:

    >> So the question is this: what is the maximum n such that "the least integer indescribable using less than n words" does not lead to a contradiction?

    This is actually a very interesting question. Why? Because this is an example of an undecidable statement (i.e. given a certain "N" you cannot proof that this is the minimum number satisfying this requirement or not).

    Exercise left to the reader… 🙂

    (Hint: http://www.cs.umaine.edu/~chaitin/lisp.html)

  11. Mr Blobby says:

    Hmmm, what about the following argument:

    English has a finite (though incredibly large) dictionary, even counting all derivative words. So we can (in principle) enumerate all phrases of length 1, length 2, length 3, etc.

    So far we’re just treating these things as symbols for arbitrary objects. Only THEN do we begin to interpret them. We have to keep ‘symbol space’ and ‘interpretation space’ separate in our minds.

    So say if we found an example of an "contradictory" phrase of length 5. If we went through the entire list of phrases of length 4 and couldn’t find a contradictory phrase amongst them when interpreted, this surely means n=5 would be the answer. And we already know n<=10…

  12. AdiOltean says:

    >> If we went through the entire list of phrases of length 4 and couldn’t find a contradictory phrase amongst them when interpreted

    That’s the problem right there. The core issue is that you can have undecidable statements in English. (for example mathematical results that cannot be proofed one way or the other).

    This means that you have no way to verify that certain phrases are contradictory or not.

    Which suggests (but doesn’t proof, I agree) that there is no way to find the minimal contradictory phrase. So, if the size of the minimal undecidable phrase is, say, 5, then how would you proof this?

    Very interesting topic, BTW!