[FOM] Foreman's preface to HST

[joeshipman@aol.com]

Your examples are good ones and delineate a boundary.

Although those statements certainly cannot be settled without higher 
set theory, they do not have much relevance to ordinary mathematics 
(most mathematicans would not regard the projective hierarchy as 
"everyday sets").  The Axiom of Determinacy is simply false if you 
accept AC, and its weaker but legal sibling, Projective Determinacy, 
is, like the statement about projective sets being Lebesgue measurable, 
of limited interest outside set theory ("descriptive set theory" is 
technically a branch of analysis but not one that influences mainstream 
analysis much).

SCH fails at aleph_omega is a nice simple statement about cardinal 
arithmetic with very high consistency strength, but unlike CH has 
little importance outside of the logic/set theory/foundations area. I 
am not sure that this will always be true; for now, though, the impact 
on mainstream mathematics of large cardinals is very small (and the 
impact on mainstream mathematics of the larger large cardinals, above 
measurable, is microscopic).

-- JS

-----Original Message-----
From: Monroe Eskew <meskew at math.uci.edu>

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.