Some programmers don’t see much relevance in theoretical CS classes (especially my students). Here is something I find very relevant. Let me build it up in pieces for those that haven’t seen it before…
A) Programming problems can be reworded to be questions about languages.
B) Turing machines recognize languages.
C) Turing machines can be encoded as (large) integers.
D) Therefore, the number of possible Turing machines are countably infinite
E) The power set of a set is just all the possible subsets of that set.
F) If a set is countably infinite, its power set is bigger, ie, uncountably infinite.
G) Therefore, if a language is infinite, it has an uncountably infinite number of subsets. Each of these represents a problem. But there are only countably many Turing machines with which to solve those problems. And if we cannot solve a problem with a Turing machine, it cannot be solved.
Conclusion…we can only solve an infinitesmally small fraction of all problems.
My question is almost here…
Whenever I present this argument to students, they get stuck on the countably vs. uncountably infinite. They generally do not have strong math backgrounds, so attempts to explain via Cantor’s diagonalization argument tends to glaze their eyes.
Usually I try to give them somthing they can grasp, like this…place a finite box over any portion of the counting number line, and we capture a finite quantity of those numbers…but place a finite box over any portion of the real number line, and we capture an infinite quantity of real numbers. A sort of evidence that there ARE more real numbers than there are counting numbers.
Finally my question…How do YOU explain the concept of multiple levels of infinity to those that have never heard of the concept, and may not be mathematically inclined?
Final Edit: I learned a lot by asking this question and I appreciate the feedback. I wasted far too much time trying to figure out what “Community wiki” actually was. I learned there is an inherent bias in some people against theory questions that I feel is simply a mistake because so much of what we do today was theory yesterday. But this bias is natural and while I disagree with them on the value of theory, I have no problem with it, and it helps me understand where my students are coming from. I do think the BS comment was unnecessary.
I do not feel this question was a poll or a preditions-for-2009 question at all. Those of you that only want coding questions with coding answers might want to re-examine that requirement. I have moved this question to community wiki but strongly feel I was compelled to do so by improper use of force.
I think your explanation is the simplest, as that is what I learned. It’s almost as if real numbers have multiple dimensions of infinity. It is infinite in one direction, but also in another.
Diagonalization is a very cool experiment, but I can see how it may go over beginners heads. It does make sense though, if it is demonstrated in a very deliberate way, going very slowly. Just throwing up numbers quick can be hard to follow I imagine.
I think the principle of Cardinality of the Continuum is also helpful, although perhaps can be simplified to a beginner level. Showing that there is more beyond simple real vs. integers can potentially help something to ‘click’.