[FOM] 0# and conservative new axioms

[martdowd at aol.com]

The principle that inaccessible cardinals exist is a conservative one.  It
extends ZFC in an extremely conservative manner, and also in an extremely
compelling one.  There is little doubt that ZFC is consistent.  That
inaccessible cardinals exist is only a slightly stronger statement:
Second order versions of ZFC have models which occur at stages of the
cumulative hierarchy.  Shoenfield's article in the Handbook of
Mathematical Logic mentions that the existence of inaccessible cardinals
can be justified by slightly strengthening the methods he uses to justify
the other axioms.                                                               

This principle is an example of the "extendability" of the cumulative
hierarchy.  A universe V containing no inaccessible cardinal can be
extended, resulting in a universe which does contain one, indeed such a
stage of the cumulative hierarchy.  Continuing, there is no largest
inaccessible cardinal.  The author has pursued this method in a series of
papers, the latest of which, "Reflective Well-Founded Relations",
states the "axiom of extensibility".

This is a conservative new axiom, which provides a baseline theory of the
extendability of the cumulative hierarchy.  Other proposed new axioms
should be evaluated, using this perspective as a criterion.  In particular,
the existence of a measurable cardinal, which is among the weaker
statements implying 0#, should be so evaluated.  The axiom of extensibility
provides a "construction" of a chain of stationary sets, and of the filter
generated by them.

By comparison, the assumption of an ultrafilter of stationary sets is
arbitrary, and no arguments of this type in favor of such have been given.
Further, the consequence that V is not equal to L can be seen as an
indication of pathology.

- Martin Dowd

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