[FOM] Two senses of generalization

[MartDowd at aol.com]

In a message dated 9/13/2012 8:33:23 A.M. Pacific Daylight Time,  
colin.mclarty at case.edu writes:

But  should we really
call this "generalization" of the theorem or should we  call it
something else?
Once small large cardinals are admitted for logical reasons, mathematical  
structures of a wide variety may be defined, and the proof of many 
properties  automatically "generalizes".  Perhaps it might be descriptive to call  
structures whose domain is an element of $V_{\kappa}$ where $\kappa$ is the  
smallest inaccessible cardinal, "small" or "ordinary".  Structures whose  
domain has rank $\kappa$ or more could be called "large".  Their existence  is 
a price paid for extending the cumulative hierarchy, although smaller ones,  
such as categories of fonite height over $\kappa$, are useful in some  
discussions.
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