This is a manuscript by the late Edward Nelson from three years ago which outlines his claim that Peano Arithmetic is an inconsistent theory. This claim was shown to be faulty , and Nelson withdrew his claim. However, I think it is interesting to look over this and see a) what ideas he thought could attack a system that is generally held to be consistent and b) where they fail (see ).
In any case, the ideas of complexity due to Chaitin and others seem to be appropriate to capture the idea of ultrafinitism, that large numbers are somehow too large
. I'm curious to see if anything workable can be extracted from this outline.
 https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html#c039553 #mathematics #ultrafinitism