TITLE: Did Edward Nelson accept the incompleteness theorems?
QUESTION [13 upvotes]: Edward Nelson advocated weak versions of arithmetic (called predicative arithmetic) that couldn't prove the totality of exponentiation. Since his theory extends Robinson arithmetic, the incompleteness theorems apply to it. But if the incompleteness theorems are proven in theories stronger than those he accepts, he could presumably reject them. So my questions are first, did Nelson doubt either of the incompleteness theorems? And second, can the incompleteness theorems be proved in weak systems of arithmetic that don't prove the totality of exponentiation?
The closest thing I can find to an answer is an excerpt from his book Predicative Arithmetic, in which he says on page 81 "at least one of these two pillars of finitary mathematical logic, the Hilbert-Ackermann Consistency Theorem and Gödel's Second Theorem, makes an appeal to impredicative concepts."

REPLY [5 votes]: Your second question has been properly answered by Emil Jerabek, I would say.  Reading some of the comments, I feel I should write the following about your first question:
From talking to Ed Nelson and to people who knew him well, I can say that Ed Nelson has for a long time been firmly convinced that the exponential function somehow leads to inconsistency (and therefore PA is inconsistent).  He has written about this at length and has pointed out some motivation for this view, like the Bellantoni-Cook characterisation of function complexity and his writings on predicativity.
Ed Nelson's deeper motivation for his view seems to have been the following:  he had a feeling that somehow fixed point constructions (like an enumeration of all partial recursive functions or Goedel's incompleteness theorems) could be 'internalised' or 'made total' to produce a contradiction like '0=1'.  Such a contradiction would only be possible given the exponential function.  At the most fundamental level, Ed Nelson did not believe that the notion of completed infinite set was formally consistent.