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

[gustavolacerda.livejournal.com]

I mean concrete enough statements.

I consider Goldbach concrete because there either there exists a counterexample or there doesn't (even though there are an infinite number of cases), and if it does it's simple to check it. In other words, either it is true or it isn't, independent of which foundational system you choose. And I would expect any foundational system to decide all statements like this.

Statements like CH, though, talk about much more abstract objects, whose "existence" I consider doubtful, for example the set of the so-called "real" numbers.

By "existence", I don't mean actual existence, but rather "concreteness". My definitions are circular, since my ideas on this are still philosophical.

I guess what I want to find is a "decidable fragment" of arithmetic, which would include concrete things like Goldbach, but not necessarily things like CH.

Do you see what I'm getting at?

[bram.livejournal.com]

If I see correctly what you are getting at, what you really want is a retraction of Godel's Theorem! Which I don't think is going to happen.

[gustavolacerda.livejournal.com]

do you know Presburger Arithmetic? I wonder how far we extend it before being forced into undecidability.
My hope is that there exists a decidable arithmetic which can express statements of the form of the Goldbach Conjecture.

[gustavolacerda.livejournal.com]

here's the correct link: http://en.wikipedia.org/wiki/Presburger_arithmetic

[bram.livejournal.com]

Thanks. I had never heard of this before. By making your system weaker you get around Godel's theorem but, well, you make your system weaker.