TITLE: Are there logical systems where formal proofs are not computer verifiable?
QUESTION [7 upvotes]: In a set-theoretic system using first-order logic, every proof could be written as a goal followed by a finite sequence of sentence where each one is justified by an axiom or previously established sentence and the last line is the goal. Computers can easily verify proofs like these using the rules of first-order logic.
In dependent type theory, proofs come in the form of programs, and you know your proof is correct if it can be compiled (or type-checked).
Do there exist logics/foundations where rigorous proofs can not be checked by computers? Or where a checking program might never terminate for a correct proof?

REPLY [8 votes]: If you take "proof" as a convincing argument (to error probability smaller than arbitrary epsilon) that a proposition is true, then two other possibilities come to mind:

Probabilistically checkable proofs, where you have a conventional first-order proof (like your secret unpublished proof of RH) and you can convince me through a cryptographic-like protocol that you really do have a proof, without revealing how the proof works.  Zero-knowledge proofs are related to this; and
(Due to L. Levin) if you believe there are physically realizable ways of generating algorithmically random numbers (e.g. by rolling dice), you can make almost-certainly-true but unprovable statements, like that the string produced by 1000 bits of coin flips can't be compressed to less than half its original length (this is an assertion about the Kolmogorov complexity, which is uncomputable).  

It follows from the second example above that the physics assertion that you can generate random numbers through quantum processes is scientifically unverifiable, if you believe the Church-Turing thesis.