[FOM] infinite logical derivations

[Ron Rood]

Dear FOM-ers,

In a Hilbert or Gentzen-style logic, logical derivations are usually 
represented as finite sequences of sentences.

Are there studies of logics in the style of Hilbert or Gentzen (natural 
deduction) which, in one way or another, allow for logical derivations 
of infinite length? If yes, can you provide me with pointers to the 
literature?

I know that tableaux systems, for example, may allow for infinite 
derivation trees. But this is not exactly what I am looking for. I do 
not have an exact positive conception of infinite logical derivation in 
a Hilbert or Gentzen system in mind. Hope my question is nevertheless 
reasonably clear.

Thanks and best regards,

Ron