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."
Best Answer
Gödel’s second incompleteness theorem requires neither exponentiation nor “impredicative concepts”. The systems Nelson works in are fragments of arithmetic interpretable on definable cuts in $Q$; one such fragment is the bounded arithmetic $I\Delta_0+\Omega_1$ (this appears to be what Nelson calls $Q_4$ in the Predicative arithmetic book). The theory $I\Delta_0+\Omega_1$ (and even weak fragments of it with more restricted induction, such as $PV_1$) is perfectly capable of proving the second incompleteness theorem (for theories with a polynomial-time set of axioms, which is not a real constraint).