[FOM] What do you lose if you ditch Powerset? (fwd)

Thomas Forster T.Forster at dpmms.cam.ac.uk
Sat Nov 15 13:06:06 EST 2003

---------- Forwarded message ----------
Date: Sat, 15 Nov 2003 12:08:32 +0000 (GMT)
From: Thomas Forster <tf at dpmms.cam.ac.uk>
To: sambin at math.unipd.it
Cc: tchow at alum.mit.edu, fom at cs.nyu.edu
Subject: Re: [FOM] What do you lose if you ditch Powerset?

There is a precedent for thinking about this.  There is the view that
power set is a very impredicative axiom and should be regarded as
implausible.  I remember being very struck, when i was a (philosophy!)
student, by the passage i read in the then-newly-minted ``Set Theory
and the Continuum Hypothesis'' by yer man;

  ``A point of view which the author feels may eventually come to be
accrepted is that CH is *obviously* false.  The main reason one accepts
the axiom of infinity is probably that we feel it absurd to think that the
process of adding only one set at a time can exhaust the entire universe.
Now $\aleph_1$ is the set [{\sl sic}] of countable ordinals and this is
merely a special and the simplest way of generating a higher cardinal.
The set C is, in contrast, generated by a totally new and more powerful
principle, namely the power set axiom. It is unreasonable to expect that
any description of a larger cardinal from ideas deriving from the
replacement axiom can ever reach C....''

  (p 151)

As i reread this, i realise that it doesn't say that the power set axiom
is implausible, but it does flag it as something that requires care.

For a number of years - possibly beco's of having this at the back of my
mind - i've been interested in weakenings of the power set axiom.  Clearly
one wants a version of it that does give *some* new sets, but not as many
as power set does.  There is a theorem of Tarski that sez that every set
has more wellorderable subsets than members so perhaps one should try
weakening the power set axiom to the assertion that the set of
wellorderable subsets of any set exists.  Tbe JSL has been so good as to
publish an article by John Truss and me on this in the current number.

I would be interested to know what other listmembers have to say about
such weakenings of ZF. 

     Thomas Forster

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