[FOM] From theorems of infinity to axioms of infinity

[MartDowd at aol.com]

Nik,
 
As I said, I'm no expert, but I recall that there has been interest in  
formalizing fragments of analysis in first order arithmetic.  Of  course the 
contortions involved make this a subject of quite limited  interest.  One 
reference concerning related subjects is
_www.andrew.cmu.edu/user/avigad/Papers/elementary.pdf_ 
(http://www.andrew.cmu.edu/user/avigad/Papers/elementary.pdf) 
 
Regarding the power set axiom, while it is true that classical mathematics  
can be carried out in the first few levels of the cumulative hierarchy, 
calling  set theory a "metaphysical extravagance " seems extreme.  The main 
uses of  higher levels of the cumulative hierarchy are in set theory itself.  
This  has advanced far beyond its origins as a tool for making analysis  
rigorous.
 
- Martin Dowd
 
 
In a message dated 3/15/2013 12:53:18 P.M. Pacific Daylight Time,  
nweaver at math.wustl.edu writes:

This is  about the worst example you could use to make this point.  Unless
your  formal system takes the real numbers as primitive, you will need some
kind  of coding machinery 
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