[FOM] CH versus the exponential function

[Sam Sanders]

Dear FOM,

Michael Finney wrote the following:

> I would also argue that people of good will can and are in permanent
> disagreement about "facts”.

In evidence of this claim, a student of the late Ed Nelson (named J. Hook)
developed mathematics inside Nelson's predicative arithmetic plus the axiom

“the exponential function is not total”. (*)

(See p. 84 of Hook’s dissertation and onwards)

It goes without saying that most of us believe the exponentiation 
to be total, while Ed Nelson did not.  Hook developed mathematics
based on this idea and (*), i.e. CH is definitely not the first milestone
where people might reasonably disagree about “fact vs opinion”.

Best,

Sam

PS: While (*) may seem strange, Ed Nelson used to mention
the Belantoni-Cook characterisation of polynomial time functions
as a motivation for (*). From this point of view, the exponential
is indeed different in kind. Nelson’s criticism of the exponential
function is therefore based on reasonable arguments, but most
of us would not jump to the conclusion of adopting (*). 

PPS: If memory serves, Peter Koelner has publicly referred to 
the totality of the exponential function as a "kind of large cardinal axiom”.  
This would have been around 2013-2014.