[FOM] Cohen was right

[Rafi Shalom]

The philosophical discussion about the ideas that motivate the powerset and
replacement is not new. The powerset axiom is easily derivable from stage or
rank theories (that make ranks of sets or equivalent notions an explicit
part of the formalization), and replacement isn't. In his "iteration again"
Boolos shows that replacement can be derived from a "limitation of size"
principle, but he thinks that such a principle is less natural (and more
paradox mindful) than the ones that motivate cumulative hierarchies.

I have a detailed discussion of this issue at http://arxiv.org/abs/1107.3519


Regarding Cohen's argument - taking the collection of all countable ordinals
may seem like a weak assertion, but its philosophical generalization is
obviously very strong and even paradoxical, so it's "weakness" depends much
on ones POV. Eventually it must have something to do with the way one
perceives sets and handle the paradoxes. In the same article there's a
unified way to handle such a generalized principle that includes cases in
which the resulting set complies with the property that established it (A
version with a better explanation of this concept is due shortly).

Best,

Rafi.

On Wed, Sep 14, 2011 at 12:06 AM, Monroe Eskew <meskew at math.uci.edu> wrote:

> On the other hand we have models like V_{\omega*2} in which powerset
> holds and instances of replacement fail and there is no set of all
> countable ordinals. But a lot of ordinary mathematics can be done
> within it, and equivalents of cardinal arithmetic can be stated within
> the model.  What philosophical lessons should be drawn from it?
> Perhaps replacement is a powerful new principle transcending powerset.
>  Or maybe we should say both axioms come from a common idea.
>
> Best,
> Monroe
>
>
>
> On Tue, Sep 13, 2011 at 11:00 AM, Ali Enayat <ali.enayat at gmail.com> wrote:
> > The following two examples justify Cohen's position challenged by
> > Monore Eskew's recent postings.
> >
> > In particular, the first ones addresses Eskew's comment that he sees
> > no philosophical difference between "completed R" (set of real
> > numbers) and "completed \omega_1." (set of countable ordinal), while
> > the second one shows the fundamental difference between "completed R"
> > and "completed alephs of all orders".
> >
> > Example 1:
> >
> > Let N be a model of ZFC in which the continuum is aleph_2; Cohen
> > showed us how to build N assuming Con(ZF).
> >
> > Let M be H(aleph_2) as computed within M, i.e., M is the collection of
> > sets that are *hereditarily* of cardinality at most that aleph_1, as
> > viewed in N,
> >
> > Then we have (1)-(3) below:
> >
> > (1) All of the axioms of ZFC with the exception of the power set axiom
> > hold in M;
> >
> > (2) The collection of real numbers DO NOT form a set in M;
> >
> > (3) The collection of countable ordinals DO form a set in M (and they
> > are the last aleph in M).
> >
> > So in M, "completed R" does not exist, but "completed omega_1" exists;
> > hence illustrating Cohen's claim.
> >
> > Example 2:
> >
> > Assuming Con(ZF + there exists an inacccessible cardinal), there is a
> > model N* of ZFC in which the continuum is a regular limit cardinal
> > (i.e., a weakly inaccessible cardinal). This is a consequence of
> > Solovay's classical modificaion of Cohen's argument in his "The
> > continuum can be anything it ought to be" paper, in which he
> > demonstarted that the continuum can be arranged to be any prescribed
> > aleph of uncountable cofinality in a cofinality-preserving generic
> > extension of the universe (Easton, in turn, generalized Solovay's
> > theorem, but that's a different story).
> >
> > In such a model N*, if we define M* as H(continuum), i.e., then we have:
> >
> > (1*) All of the axioms of ZFC with the exception of the power set
> > axiom hold in M*;
> >
> > (2*) The collection of real numbers DO NOT form a set in M*';
> >
> > (3') There is no last aleph in M*.
> >
> >
> > Regards,
> >
> > Ali Enayat
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> >
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