On the consistency strength of the inner model hypothesis
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by
Sy-David Friedman, Philip Welch, W. Hugh Woodin
2008 Volume 73, Issue 02, p391-400
Abstract
The <jats:italic>Inner Model Hypothesis</jats:italic> (IMH) and the <jats:italic>Strong Inner Model Hypothesis</jats:italic> (SIMH) were introduced in [4]. In this article we establish some upper and lower bounds for their consistency strength.
We repeat the statement of the IMH, as presented in [4]. A sentence in the language of set theory is <jats:italic>internally consistent</jats:italic> iff it holds in some (not necessarily proper) inner model. The meaning of internal consistency depends on what inner models exist: If we enlarge the universe, it is possible that more statements become internally consistent. The <jats:italic>Inner Model Hypothesis</jats:italic> asserts that the universe has been maximised with respect to internal consistency:
<jats:italic>The Inner Model Hypothesis</jats:italic> (IMH): If a statement <jats:italic>φ</jats:italic> without parameters holds in an inner model of some outer model of <jats:italic>V</jats:italic> (i.e., in some model compatible with <jats:italic>V</jats:italic>), then it already holds in some inner model of <jats:italic>V</jats:italic>.
Equivalently: If <jats:italic>φ</jats:italic> is internally consistent in some outer model of <jats:italic>V</jats:italic> then it is already internally consistent in <jats:italic>V</jats:italic>. This is formalised as follows. Regard <jats:italic>V</jats:italic> as a countable model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that <jats:italic>V</jats:italic>* is another such model, with the same ordinals as <jats:italic>V</jats:italic>. Then <jats:italic>V</jats:italic>* is an <jats:italic>outer model of V (V</jats:italic> is an <jats:italic>inner model of V</jats:italic>*) iff the sets of <jats:italic>V</jats:italic>* include the sets of <jats:italic>V</jats:italic> and the classes of <jats:italic>V</jats:italic>* include the classes of <jats:italic>V</jats:italic>. <jats:italic>V</jats:italic>* is <jats:italic>compatible with V</jats:italic> iff <jats:italic>V</jats:italic> and <jats:italic>V</jats:italic>* have a common outer model.
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