more ignorant logic speculation

[gusl]

So, I've been thinking, how significan't should Gödel's Incompleteness be anyway? It seems those example sentences are always pretty contrived, whereas I only really care about "concrete" statements.

Give me an example of a concrete-looking undecidable statement in number theory.

Should we replace the ideal of static axiomatization with a dynamic one? Can there be an algorithmic way of picking new axioms? Would this create a logic on its own, which also suffers from Gödel's Incompleteness?

Chaitin views Gödel's incompleteness as an information-theoretic necessity.


Does the undecidability of FOL imply that there are no bounds on proof size?

Comments

[spoonless.livejournal.com]

Indeed... Godel's Incompleteness Theorem only applies to systems with "second order logic"... systems which can make statements not only about their elements, but also about sets of those elements.

It seems to me that any statement about a finite set can be expanded out as a finite sequence of statements about the elements of the set. In which case, a system is only irreducibly second order if it makes statements about infinite sets. (I'm inserting my own word "irriducibly" here just to mean, "not reducible to first order").

Then again, I've never actually seen this claimed anywhere, so maybe there are some pathological cases where my reasoning isn't right. But it seems like a reasonably solid argument.

[spoonless.livejournal.com]

uh... sorry about that. Guess I have no idea what I'm talking about here. :(