# [FOM] latest new axiom

martdowd at aol.com martdowd at aol.com
Mon Nov 30 16:00:09 EST 2015

Chains of sets of inaccessible cardinals can be constructed, so that as long as
the sets remain stationary the chain is strictly descending.  New axioms for
set theory may be given, by postulate that there are cardinals where the chain
remains stationary.  The simplest example is, that Mahlo cardinals exist (i.e.,
the inaccessible cardinals form a stationary set).

These axioms can be justified as specifying "endlessness" properties of the
cumulative hierarchy, and it is certainly of interest what cardinals can be
"built-up" in this fashion.  Whether a weakly compact cardinal can be built-up
is a paramount question.

In http://www.hyperonsoft.com/s11.pdf the axiom is proposed, that there are
cardinals $\kappa$ where there are stationary set chains of length $\alpha$ for
any ordinal $\alpha$ which is the order type of a WPS (well-order on a subset)
$\preceq$ on $V_{\kappa}$ such that $\preceq$ has a $\Delta^1_\infty$
definition which is uniform for all inaccessible cardinals $\lambda\leq\kappa$.

In a new paper http://www.hyperonsoft.com/cohk.pdf the axiom is proposed, which
requires uniformity of a well-founded relation
on a club of cardinals.  This axiom has some technical advantages
over the previous one.

Let $\Omega(\Sigma^1_1)$ denote the supremum of the order types of the
$\Sigma^1_1$ WPS's.  In http://www.hyperonsoft.com/wc.pdf it is claimed that if
$V=L$ then the order types of the WPS's which are uniformly $\Sigma^1_1$ in
inaccessibles equal $\Omega(\Sigma^1_1)$.  I reported this in an earlier
posting to FOM.  The proof is incorrect, however, and results in the new paper
suggest that this may not be true.

There is a gap'' between ${\cal R}\Sigma^1_1$-Mahlo cardinals
and weakly compact cardinals.  Closing this gap provides a specific
method for attempting to build up a weakly compact cardinal.

Martin Dowd
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