Note on proofs of inconsistency
There are many discussions going on around the web in the wake of E. Nelson’s claim about inconsistency of PA . While I can’t contribute on a professional level, I’d like to make a trivial note:
Nelson is not trying to prove the statement “the theory is inconsistent” within the theory itself. Such a proof would not have any consequences regarding actual consistency of the theory.
I wrote about this more than once already: there are consistent theories that prove own inconsistency . The reason being that consistency is not a very strong condition. Consistent theories may potentially lie. Consistent theories cannot prove Π-falsehoods, but they can prove Σ-falsehoods. I outlined the relationships between consistency and soundness here.
Now, what Nelson is doing is really straightforward. He is simply deriving a contradiction within the theory. Not something like “0=1” of course (though if he is right, then a proof of “0=1” does indeed exist).
Stay tuned, to be continued…