[FOM] Two senses of generalization
MartDowd at aol.com
MartDowd at aol.com
Thu Sep 13 11:57:55 EDT 2012
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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