[FOM] Improving Set Theory

[martdowd at aol.com]

 FOM:                                                                        

In response to Harvey Friedman's post on improving ZFC, at the risk of
being repetitive, I have a series of papers on adding "conservative" new
axioms.  The first is "Some new axioms for set theory"                          
 http://www.ijpam.eu/contents/2011-68-4/4/4.pdf
The most recent is "Reflective well-founded relations"
 http://www.ijpam.eu/contents/2016-107-4/6/6.pdf

The "conservative" new axiom approach proceeds from the observation that
the existence of an inaccessible cardinal is a "semantic" version of
the consistency of ZFC.  $V$ can be "collected" in to a set, namely,
$V_\kappa$ where $\kappa$ is the smallest inaccessible cardinal.
The first paper argues that this process can be continued to a Mahlo
cardinal.  The second argues that the existence of cardinals satisfying
the "axiom of extensibility" Ax_{Ext} is strongly justified.

These cardinals are stationary below a weakly compact cardinal.  In my
opinion whether the existence of a weakly compact cardinal can be
justified "from below" by conservatively extending the universe is an
important one.  I have been working on other subjects, and have left the
situation as of the second paper.

- Martin Dowd

 
 
-----Original Message-----
From: Harvey Friedman <hmflogic at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Thu, Jan 9, 2020 4:54 pm
Subject: [FOM] Improving Set Theory

ZFC has become the standard foundation for mathematics since about
1920. Alternatives have been proposed but not widely endorsed at least
not yet.

I am particularly interested in what people think is lacking or is
flawed about ZFC.

What needs exploring is to what extent and how can ZFC can be
adjusted, modified, improved to meet such objections or requirements.

My general thesis is that ZFC is extremely flexible and supports many
modifications in many different directions for may different purposes,
and that such modifications or adjustments of ZFC are far preferable
to any kind of overhaul in f.o.m.

Harvey Friedman
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