Every Number Is Special In Its Own Special Way

Every Number Is Special In Its Own Special Way

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I got a question recently about where in the .NET framework the "special numbers" were defined. The questioner was actually asking about the Double.NaN, Double.PositiveInfinity, etc, special values for floating point numbers. Of course there are other "special numbers" defined by the framework, such as Math.PI.

The question was easily answered but it got me thinking, which, as we know, is usually trouble.

Clearly zero is a very special number, being the first natural number.

One is pretty special too, being the multiplicative identity.

Two is the only even prime.

Three is the lowest odd prime...

Clearly lots of numbers are special. This led me to propose the following theorem:

Theorem: Every natural number (0, 1, 2, ...) is a special number.

Proof:

Let’s posit that the set of nonspecial natural numbers is nonempty, and deduce a contradiction.

If there exists a nonspecial natural number then there must be a lowest nonspecial natural number.

What an unusual property! The lowest nonspecial natural number!

Whatever number has that unusual property must be kinda... special.

Therefore the lowest nonspecial natural number is special.

Therefore, if the set of nonspecial natural numbers is nonempty then it contains a special number.

That is clearly nonsensical, therefore the set of nonspecial natural numbers is empty.

Therefore all natural numbers are special, QED.

And yet I can't find anything special about 7920687935872092847630945767548023. But it must be special somehow!

Extending the proof show that all real numbers are special is left as an exercise for the reader.

  • Ah, but what about "The lowest positive integer that cannot be uniquely identified in twelve words"?

    Your proof is flawed for the same reason that sentence is. If only I could figure out what that reason *is*...

  • Indeed, I am really making no proof at all. Both "uniquely identified" and "special" are such ill-defined predicates that all we are really proving is that you can't make equivalence classes with such lousy predicates!

  • 7920687935872092847630945767548023 is the only number that, when you subtract 1 from it, yeilds 7920687935872092847630945767548022!

  • You haven't defined "Special".  Perhaps none of these numbers are "special". Is it merely having an "unusual property"?

    ;)

  • I think the flaw comes from naive set theory (http://en.wikipedia.org/wiki/Naive_set_theory). Basically, the contradiction is a restatement of the well known paradox, Russell's paradox: "the set of all sets that do not contain themselves as members". I think you need to move over to axiomatic set theory (http://en.wikipedia.org/wiki/Axiomatic_set_theory) if you want a sounder proof... :)

  • Correct.  There are two errors here.  First, I haven't actually defined the predicate "special", and second, I'm making the possibly unwarrented assumption that a predicate defines a set.

  • Furthermore, I imagine that extending this "proof" to the real numbers can turn out to be a real pain, as we rely on ordering/enumerating all the special numbers, the real numbers being unenumerable and all...

  • Are we really basing it on enumerating the numbers, or just that there exists a lowest? It looks to me like it works fine with reals, since if there exists a nonempty set of nonspecial real numbers, aren't we guaranteed that one of them is the lowest? Aren't the real numbers completely ordered? Or is there some math principle I'm missing here?

  • No, absolutely we are not guaranteed any such thing!  That's not even true of nonempty sets of integers.  What's the lowest of {-1, -2, -3, ... }  ?  There isn't any.  Now, of those you can pick a different definition of "lowest" like "lowest in absolute value."  That always works for sets of integers.

    But what about fractions?  Consider {1, 1/10, 1/100, 1/1000 ... }  There's a bounded set with no lowest value in the set. and we can't even do some trick like "closest to zero".

    And the reals are NOT enumerable.  Consider if there were a complete enumeration and deduce a contradiction: construct the sequence of zeros and ones such that the nth element of the sequence is one if the nth digit after the decimal place of the nth real is zero.  The nth element of the sequence is otherwise zero.  Now construct the real number consisting of 0. followed by all those zeros and ones.  That thing is a real number.  Is it on the list?  Which place could it be in?  It differs from every element in the enumeration by at least one decimal place, so it can't be equal to any of them, therefore our supposition that the enumeration is complete is false.

    Do a web search for "Cantor's diagonall argument" for more details.

  • I believe your notion of "special" here is axiomatizable -- indeed, it's something I've been thinking about for a good few years. Suppose we have a language of symbols, and an interpretation which maps finite sequences of symbols into set theoretic predicates. We say that a set is expressible (in your terminology "special") if there exists a finite sequence of symbols (an expression) which maps to a predicate which holds for that set and no others.

    Every natural is trivially expressible (maybe it's "the only natural number which, when expressed in base 10, goes 1-7-4-3-3-9-2-4"). But back on to ordering and the reals. While you point out that the reals are not enumerable, they are well-orderable (if we assume the axiom of choice). However, while the set of all well-orderings of the reals is non-empty, it has no expressible members (for any sufficiently powerful language).

    Proof: In any language L, there are only at most a countably infinite number of expressible sets. Since the reals are uncountable, the set of non-expressible reals is nonempty. Therefore, if we can express a well-ordering of the reals, and we can express the notion of expressibility-in-L within L, then we can express the least non-expressible real, a contradiction. []

    Another neat property is that, for any class of languages where we can express a transformation from any language in the class to any other, in some language in the class, the classes of expressibles within all those languages are in fact the same class! In layman's terms, switching from English to French doesn't allow us to talk about anything new.

    Finally, the expressible subset of the complex numbers is algebraically closed, countable, and contains every complex number anyone will ever need. :p

  • In this case, a predicate does define a set.  In axiomatic set theory, there's an axiom that lets you extract a subset of a larger set by filtering it with a predicate.  In your argument, you're assuming at the outset that all predicates will be applied to the integers.  That's legal.  What's illegal is to use a predicate to construct a set that is not a priori a subset of something else.  That's what would get you into the Russell paradox.

    Your theorem is usually known as the "smallest uninteresting number paradox".  I once saw a talk by a mathematician who decided to test the statement.  He began as you did, pointing out interesting facts about counting numbers in order.  When got to eleven, he thought for a minute, shrugged his shoulders, and triumphantly declared eleven to be the smallest uninteresting number.  The rest of the talk consisted of showing us all the ways that eleven was boring.

  • http://en.wikipedia.org/wiki/Interesting_number_paradox

  • I first saw this proof (I think) in The Penguin Dictionary of Curious and Interesting Numbers:

    http://www.amazon.com/Penguin-Book-Curious-Interesting-Numbers/dp/0140261494

    I haven't looked at it for 20 years, but I remember enjoying it immensely.  The geometry one's even better:

    http://www.amazon.com/Penguin-Dictionary-Curious-Interesting-Geometry/dp/0140118136

  • So on Tuesday, Eric Lippert posted about how Every Number Is Special In Its Own Special Way . Now everyone

  • Coming from the Liberal Arts, I think every number is special in its own way, and I think every number should be awarded a handsome Certificate of Participation suitable for framing.

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