# [FOM] Foreman's preface to HST

Monroe Eskew meskew at math.uci.edu
Tue Apr 27 00:45:03 EDT 2010

On Sat, Apr 24, 2010 at 6:30 PM,  <joeshipman at aol.com> wrote:
>
> Secondly, most of the natural independence-from-ZFC statements in the
> non-Absolute part of mathematics, such as the Continuum Hypothesis,
> have no additional consistency strength and can be shown independent by
> forcing arguments that can be formulated syntactically or in a much
> smaller set-theoretic universe that does not essentially depend on "the
> higher infinite".

I'm not sure what you mean by "natural," but there are many statements
about everyday sets that have pretty high consistency strength.  A few
examples:

1) The Axiom of Determinacy.
2) SCH fails at \aleph_{\omega}.
3) All projective sets are Lebesgue measurable.