I recently read this pdf: Warning Signs of a Possible Collapse of Contemporary Mathematics, and I'm having some trouble understanding the issues it raises.
The author says that the consistency of Peano Arithmetic cannot be proved within the system, which is a direct application of Gödel's second incompleteness theorem. It can, however, be proved using transfinite induction, but this is a problem in his opinion, because one has to trust the existence of infinite cardinals, contrary to his monotheistic faith. My first question is whether this is what he meant, or if I misunderstood some part of his argument.
Nelson then goes on to define some classes of numbers which are contained in one of them, that of "counting numbers", and proves that these counting numbers are closed under addition and multiplication. But he implies that it is not possible to prove that exponentiation and superexponentiation and, in general, faster-growing recursive functions, whose definition is completely permitted by the axiom of induction, are guaranteed to terminate, i.e., to be computable. My other questions are: can all recursive functions be shown to terminate after a finite number of steps, assuming the existence of some transfinite cardinals? And what exactly is the finitist objection to this assumption? Finally, what would be a finistically acceptable model for mathematics? Is it usable, in a general sense?