[FOM] From theorems of infinity to axioms of infinity

[MartDowd at aol.com]

Nik,
 
J. Shoenfield's article in the "Handbook of Mathematical Logic" contains a  
justification of the power set axiom.  Its adoption can be seen as an  
application of Occam's razor.  It is necessary to have the set of all  functions 
from the reals to the reals, even for calculus; then subsets such as  the 
continuous functions,and differentiable functions can be defined.  This  
illustrates that the power set of a set is a self-evident and useful concept;  
the simplest way to ensure its existence in formal set theory is to adopt the 
 power set axiom.  The fact that this leads to sets outside the realm of  
ordinary mathematics could be considered a metaphysical accident rather than 
an  extravagance.
 
= Martin Dowd
 
 
In a message dated 3/19/2013 5:24:01 P.M. Pacific Daylight Time,  
nweaver at math.wustl.edu writes:

The  point I have to emphasize is that I am not aware of any even
remotely  cogent justification for making this jump.  There is no
clear  philosophical reason to suppose that infinite power sets are
surveyable,  nor is there any practical reason, as it turns out that
the vast bulk of  mainstream mathematics can do perfectly well without
this assumption.   Does that help explain why I use the term  "metaphysical
extravagence"?


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