[FOM] infinite logical derivations

[Kreinovich, Vladik]

Yes, Carnap's rule (also known as Carnap rule) deduces \forall n A(n)
from the infinite number of premises A(0), A(1), ... Youc anf ind
numerous papers by Googling, I first learn about it from G. Mints'
lectures.  

It can formalized if we have a recursive infinite tree. 

> From Ron Rood

> 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?