# What can be done with computability logic that previous logic systems can't?

I've been reading a lot about computability logic lately and I'm superficially aware that it unifies classical, intuitionistic and linear logics.

What I'm seeking to know is:

Can computability logic be used as a foundation for mathematics (or rather arithmetic) in the same sense classical logic has been proposed before by Russell, Whitehead and other authors? Is it subject to the same restrictions (like Gödel theorems) like the other logics (considering that it conservatively extends classical logic)?

I hope I've been clear enough.

References:

http://www.cis.upenn.edu/~giorgi/cl.html

http://www.csc.villanova.edu/~japaridz/ICL.pdf

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I don't know anything about continuous logic. Yet, if---as you say---it is conservative over classical logic (for all first order formulas?) then wouldn't it have to satisfy some form of Gödel's incompleteness theorems? For example, one wouldn't be able to prove new theorems of arithmetic in continuous logic so arithmetic in continuous logic would still be incomplete. Also if it is conservative, couldn't one add the axioms of ZFC to continuous logic to get a foundation for math. (However, maybe I don't know what "conservative" means in this regard, as I don't know the syntax.) – Jason Rute Jun 3 '13 at 22:36
I kept saying "continuous logic" but I meant to say "computability logic". – Jason Rute Jun 3 '13 at 22:47