# [FOM] latest new axiom

MartDowd at aol.com MartDowd at aol.com
Mon Mar 3 21:04:56 EST 2014

The latest new axiom I am proposing is much stronger than those of  earlier
postings of mine to FOM.  The manuscript has been submitted, and  may
temporarily be found at http://www.hyperonsoft.com/s11.pdf.

The axiom states that for any well-order (actually any WPS, which is  almost
a well-order) on Ord, which has a definition in the second order  language
of set teory, which uniformly defines a well-order in V_lambda for  any
inaccessible cardinal lambda, Mahlo's operation may be iterated along  the
order on the inaccessible cardinals, and the resulting class will  be
stationary.  A justification of this axiom can be given, which does  not
require methods much stronger than those required for greatly  Mahlo
cardinals.

A wealer axiom only allows a Sigma^1_1 definition of the order.  If  kappa
is weakly compact then V_kappa satisfies the weaker axiom.  It is  of
considerable interest whether the stronger axiom implies the existence  of
weakly compact cardinals.

Among many questions which arise in this connection, is the question
whether
a bound on the order type of a well-order < on V_kappa which is  Sigma^1_1
definable in V_\kappa, can be bounded.  The trivial bound is  (2^kappa)^+.
It might be the case that this can be reducd to kappa^++.   I would be
greatly interested in any comments on this question.

- Martin Dowd
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