Indescribable numbers: The theorem that made me fall in love with math

But just to continue the mind-games, yes there are an infinity more indescribable numbers than there are describable ones, but should there be?

Given the balloon example, he makes the case that as an imaginary god: "...stare imploringly into the siren call of the final ellipsis, for you know that no matter how often you expand it, I will always smile when giving you more and more digits, because you and I will both know that the final wisdom of The Number will always be mine and never yours..."

But we know that, given any exact point in time, only a certain number of atoms are inside the balloon. Therefore, whatever the number is to answer the equation, it is one with a finite number of digits. (One imagines the experiment being done in such a manner that quantum tunneling is minimized)

So while yes, using symbols you can certainly construct numbers which are indescribable (given either an infinite series of symbols or a self-recursive way of generating more), does the real world actually work that way?

In other words, is mathematics truly isomorphic with reality as we observe it? Or is it somehow a superset of logic sitting above an infinite number of possible realities? In a universe full of discrete things (perhaps at the sub-quark level), do we need a system of mathematics that works over continuities? Is such a model always helpful? If not, where might it be tripping us up?

Also, do -any- numbers really exist? They are really just concepts we use to describe the world. Right before me, there are 3 conglomerations of molecules that we would describe as apples, but what does it mean to say the number '3' exists? It only really exists in our minds. Pi only describes perfect circles - which don't exist. So yes, infinite strings of digits which can't be described can still be thought about, but that doesn't bring much to the table.

Eventually he will discover, "God made the integers. All else is man made.".

In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Really his discoveries are about the completeness property of the real numbers. So, in particular, if we have an infinite series that meets the Cauchy criterion, then it converges, in particular, there is real number for it to converge to. The same statement is not true in the rational numbers or the algebraic numbers!

Calculus: Sure, the elementary properties of the completeness property of the real numbers.

How do we know that anything like the real numbers can exist? Because we can start with the simplest things, say, just the empty set, do a lot of set theory pushing around, construct something that looks like the natural numbers -- 1, 2, 3, .... Then we can use the naturals to construct (something that looks like) the integers -- ..., -3, -2, -1, 0, 1, 2, 3, ....

Continuing in this way, we can construct the rationals and the reals. For the reals, a popular approach, nicely intuitive, is Dedekind cuts.

So, we base it all on just set theory starting with just the empty set.

So, the reals exist but only because man said that they exist!

Too soon he will face a danger, compactness! That's where every infinite subset has a limit point, that is, in the infinite subset is a sequence that converges to something. This is true if and only if every open cover has a finite subcover. And every closed and bounded subset of finite dimensional, real Euclidean space is compact. A real valued continuous function with domain a compact set is uniformly continuous and bounded and achieves both its upper and lower bounds. Seeing these results, the poor guy might lose it! The usual way we show that the Riemann integral of calculus exists is via uniform continuity.

After he recovers from seeing the completeness property of the reals, under no circumstances let him learn about Hilbert space -- a complete inner product space! Okay, but are there any examples? Actually, yes: The set of all real valued random variables X so that E[X^2] is finite. Yup, the set of all of these is complete and, thus, forms a Hilbert space. Totally mind blowing that any such thing could be true! Here complete means Cauchy convergent means convergent where we consider distance in Hilbert space which is the metric we get from the inner product. So, with just that concept of distance, a Cauchy convergent sequence of E[X^2] finite random variables actually converges, that is, there is a random variable for the sequence to converge to. You'd think that random variables could wiggle too much, but, no, they can't! Beyond belief.

If he survives these severe trials of the mind, keep him away from a classic text on point set topology, e.g., Kelley. There learn that can have a set A and a point x so that point x is right next to set A but there is no sequence in set A converging to point x. That is, sequences are not enough to characterize the more general case of convergence. For this more general case, there is Moore-Smith convergence, nets, filters, etc.

Somewhere in there he will discover the continuum hypothesis, model theory, etc.

But under no circumstances let him get near

John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces', ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.

If he sees this book, he may never recover!

Related (and more precisely defined) concepts include (non)definable and (non)computable numbers, http://en.wikipedia.org/wiki/Definable_number and http://en.wikipedia.org/wiki/Computable_number respectively.