[FOM] Pluralistic Foundational Crisis?/set theory/completed

[martdowd at aol.com]

Niel Barton writes:

 


 Is there a unique, maximal, proper class universe of set theory?



 

 My research on new axioms based on the extendability of the cumulative hierarchy has raised interesting questions in this connection.  The "absolute" universe" of set theory is a highly problematical concept, even disregarding issues of pluralism.  If pluralism is rejected, the absolute univers decides first order sentences of LST.  Some such, such as CH, are decided in V_kappa for any (in particular the smallest) inaccessible cardinal.  GCH is more problematical, in that it is a statement extending throughout the entire cumulative hierarchy.  It may be seen as an aspect of the absolute universe that such statements may be made.

For another example, strong arguments can be made that inaccessible cardinals exists.  Otherwise, the cumulative hierarchy would lack a desirable extendability property, and the absolute universe would be "too small".

One view is that the absolute universe does exist, but mere logicians end up in a quandary trying to decipher it.

- Martin Dowd

 




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