Induction on proofs in infinitary logic

Question

In propositional infinitary logic over countable conjunctions, you may provide proofs of countable length. My question is how you may perform induction on proofs then and is there a last element of the proof and if so how is it determined.

Answer 1

With induction there does not need to be a last element ... think of standard forms of mathematical induction where we show things to be trus for all (i.e. Infinitely many!) natural numbers. There does need to be a 'next' element though, but all that is fine for infinitary logics where we have infinite but still countable length proofs. So, induction works perfectly for infinitary logics.