0
$\begingroup$

Where can I read a proof of this?

$\endgroup$
1
  • 1
    $\begingroup$ Without the assumption that there is a Woodin cardinal, this can fail. For example, every countable tree is weakly homogeneous via principal measures, but in $L$ there is a countable tree whose projection is not determined. (There are also some less trivial examples.) On the other hand, every homogeneous tree is determined, and a proof of this is implicit in Martin's proof of $\Pi^1_1$ determinacy from sharps as mentioned in Andres's answer below. $\endgroup$ Jun 17, 2013 at 19:23

1 Answer 1

3
$\begingroup$

Not sure whether this is needed anymore, but:

The paper you want is John Steel's "The Derived Model Theorem." This paper gives a thorough and superb presentation of weak homogeneity and much more, including the result you are asking for. It is an unpublished note; the latest version is dated May 29, 2008, and can be downloaded from his page.

However, you may want to take a look at "A proof of projective determinacy" by Martin and Steel, Journal of the American Mathematical Society, 2(1):71–125, 1989, although it may not look quite as what we are used to think of these things now.

And probably you want to read first Martin's proof of determinacy of $\Pi^1_1$ from sharps, since the key ideas are there (this should be in Jech's or Kanamori's book).

A modern, very quick and nice exposition of this and the key related ideas in proving determinacy at the projective level is in Itay Neeman's paper "Determinacy in $L({\mathbb R})$", in the Handbook of Set Theory. You can currently download the paper from Itay's page.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.