A most disturbing formula

It's hard to say exactly why I find this so disturbing. I continue to struggle with my understanding of what constitutes a real number, as can be seen in my other posting. This is connected to the idea of continuity, which has led to reading on the nowhere continuous Dirichlet's function, where f(x) is 1 if x is rational and zero otherwise. I had taken some comfort earlier in thinking this was a somewhat arbitrary distinction, but that was broken when I came across an algebraic formula for this function. It is f(x) = lim k->infinity ( lim n->infinity ( Cos( k! pi x ) ^ 2n ) ) Sorry for the poor notation - I'll have to look up better ways of typing formula in blogger. Anyway, let's take a look at this. It's a double limit of a cosine function, which itself is an infinite summation. All these infinities can make one uneasy, but they really don't do all that much. Let's look at the effect n has on the formula. The idea is that since the cosine function is bounded between zero and one, any exponentiation of its value will leave the one unchanged, but drive any value less than one closer to zero. The limiting process of that exponentiation is simply to restrict the value to either zero or one and eliminate all intermediate values. This is what introduces the discontinuity when the cosine equals 1, but otherwise the function would be continuous everywhere else. This is where the factorial of k comes in, and it's rather clever. k! assures that we can multiply out the denominator of any rational x, transforming it into an integer. That means we'll have an integer multiple of pi as the variable of our cosine, and since half the numbers in k! are even, it further means that we'l have an even integer multiple of pi, which of course assures us that the cosine produces a one at every rational number.

Since irrational numbers cannot be expressed as the ratio of integers, no integral multiplier, no matter how large, can change an irrational into an iteger. Therefore the above cosine for irrational x can never be given a multiple of 2 pi and so will always produce less than one, which the exponentiation can drive to zero. This gives us exactly Dirichlet's function where all rationals map to one and all irrationals map to zero.

The disturbing thing isn't that an irrational multiplied by infinity is still an irrational, because that's not how the limit process works. Consider any finite but large integer k multipling an irrational. The result will still be irrational. The nature of integer times irrational doesn't change just because k gets bigger. The fact that we're using k! means that k! is way bigger than k, but finite integer k gives just as finite an integer k! The limiting process doesn't change this essential characteristic. The use of lim k-> infinity of k! is an ingenious way to convert all rationals to integers, and it does that well. Note that both the integers and the rationals are of the same cardinality. They are both countably infinite, and this is somehow comforting.

(Side thought: However, nothing in the formula explicitely defines k as a sequence of integers, though it is certainly implicit in the factorial operation, which is only defined on integers. Would extending he formula to use the Gamma function change the conclusion? Certainly Gamma operating on an integer gives an integer, and so does Gamma operating on a (countably infinite) number of other values produce integers. I wonder if Gamma only produces an integer when operating on a rational. Is that even an important consideration? Let me think about that later and write more in a reply...)

Somewhat easier to see but more discomforting is the use of the limit in exponentiation. Again, it seems simple enough on the surface. For any small number, exponentation makes it smaller. The limiting process of that is zero. It strikes me as essential that zero is the identity element, because that means any modifications of a value by this element leaves the value unaltered. Essentially, there is no trace that such an operation was performed. To me this says that when zero is the result of a limiting process, continuity may be lost, because a series of increasingly untraceable operations drive the limit. This argument certainly lacks rigor, and may be deeply flawed. It's kind of at the heart of my whole difficultiy with understanding what a real number is. But I digress.

In fact, I've digressed more than enough. Since this is just a blog entry and not a paper, I'l try to collect my thoughts and make a follow up post later. In the meantime, your comments would be very appreciated.