I recently went through my approximately 3-year cycle of wondering what the significance of the Halting problem is all about and wrote up my thoughts here: http://credentiality2.blogspot.com/index.html (I include the link for completeness, but it's not the point of this message, and I can't claim that it's in any way worth the time it takes to read it.) My old CS theory prof tells me I take offense to the notion of uncomputability because I prefer the constructivist school of logic: http://en.wikipedia.org/wiki/Constructivism_(mathematics) I poked around wikipedia a bit and have to agree that it does appeal to me. In particular, I like the idea that the reals are computable and perhaps thus not 200% more infinite than the naturals as I had been lead to believe, but rather 100% less different in cardinality, there being no more fewer reals than naturals than there are naturals and integers. Excuse me, I think I need to sit down for a moment. Anyway, it seems that Constructionism has not yet converted all the heathen masses, and I'm curious what this mass of heathens thinks of it. In particular, I'm curious what it means for diagonalization proofs, which have always seemed suspicious to me, doing magical things that only work out at infinity.