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I'm seriously intrigued both by some of the ideas that emerged in conversation this morning, and by some of the social dynamics underlying that.  Some thoughts along both lines, from the session as well as conversations after.

I'm happily getting more familiar with the details of Gödel as well as the context of his work.  Glad to have Alan around this morning to help with both Godelization and the diagonalization argument first used by Cantor.  Together it seems to me they give us tools that will not only help with Turing and Chaitin but can be used more generally, as suggested in my summary above.  The distinction between the "countably infinite" and still larger infinities, and the ability to make that distinction, seems to me quite important in thinking generally about inquiry.  And interestingly related to Wittgenstein's "of that which we cannot speak we must remain silent."   Human expressions of understanding are "countably finite" since they rely on formal symbol systems, but there may exist human understandings beyond those, "inexpressible" in particular formal symbols systems used by particular people at particular times.  Rather than "remaining silent" about those, one might deliberately alter one's formal symbol system in an effort to make them expressible.  The point here is not that there isn't at any given time an "unknown" which must remain unknown but very much the opposite: that there is at any givent time an unknow/inexpressible that one might in principle reach by a change in one's language/formal system.  The limitations that become apparent when one develops formal systems becomes, on this view, an incentive to develop ways to go beyond them, to develop new formal systems, to appreciate their limitations, and so on.  A nice emergent perspective on inquiry. 

Yet to be fully clarified, in my mind, is how self-referentiality and the consistency problem relate to this.  Are they correlates of it, or additional characteristics, more relevant in some contexts, less in others?

The dynamics underlying the conversation continues to be one of concern about whether an appreciation of formal systems is helpful to those less inclined to them contrasted with concern that recognizing the limitations of formal systems is either practically irrelevant ore represents a slippery slope in the battle against irrationality.  Maybe with the legitimization of the "inexpressible" we're closer to answering the first set of concerns?  With regard to the second set, my hope is that the Turing/Chaitin discussions will help with the problem of relevance.  As for the slippery slope, I continue to feel that "risk" is not a problem unique to either position; the world has as much to fear from rationality as it does from irrationality.  And that biological evolution provides a nice model of how to get this less wrong: use both formal systems and some indeterminacy and trust the dyanimcs of interaction with external pressures to keep the whole thing under control.  "Risk" cannot be eliminated along any productive/generative path, but it can be treated as a virtue rather than a deficiency (cf Evolution/science: inverting the relationship between randomness and meaning).

The dynamics is also interesting since it reminds me of old "Two Cultures" conversations, and suggests we're still in a mode in which some scientists and some humanists seriously mistrust one another.  The origins of that mistrust have never been entirely clear to me but an interest in/distaste for "formal systems" may have something to do with it.  Formal systems seem "inhuman" and therefore threatening to some people, whereas their very formality is appealing to others.  Interestingly, both sides defend their positions in terms of contributions/threats to social well being.  I wonder whether one can't do better for the latter by making common cause rather than by treating an alternate possibility as a distinctive threat.  And to what extent the apparent "inhumanity" of formal systems is a matter of how it is talked about rather than the thing itself.