2
$\begingroup$

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe. But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe.

My question is:

Where can I find a good reference for the last fact?

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ In case it makes easier for you to find: The fact you are looking for, I believe, is credited to Mostowski (so you can search for: Mostowski's absoluteness). $\endgroup$
    – Burak
    Commented May 10, 2014 at 17:18

1 Answer 1

Reset to default
9
$\begingroup$

This is an immediate consequence of the fact that every $\Pi^1_1$ statement is equivalent to the assertion that a certain relation is well-founded, and well-foundedness is absolute between transitive models. If a larger model thinks a relation is well-founded, then the smaller model must agree since it can have no infinite descending sequence. If a smaller model thinks a relation is well-founded, then it will have an ordinal ranking function, which will witness well-foundedness in the larger model.

This theorem is contained in every descriptive set theory book, such as the excellent one of Moschovakis's.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .