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I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the well-founded part of a model? If someone could give me a precise definition (maybe it can be defined using transitive closures, but I don't really know) of the well-founded part of a model, it'd be greatly appreciated.

Addendum

The well-foundedness that I'm referring to is not the internal well-foundedness that comes from assuming the Axiom of Regularity within the model. It's an external property, as viewed from outside the model.

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    Sure! I un$d$erstand that. It's just that Brian had me for a while there.2012-10-26

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Suppose that $(M,E)$ is a model of ZFC, this is a set in the universe (which is also a model of ZFC, for our purposes).

It is possible that $(M,E)$ is not a well-founded relation. Internally, of course, this is impossible. $M$ does not have any element which is a decreasing sequence in $E$, since $M$ satisfies the axiom of regularity.

However we, as educated men staring at $M$ externally, know that it is possible that $M$ has more than it knows about. One can now ask about the ordinals of $M$. Namely $(Ord^M,E)$ as a linear order. This order has a maximal initial segment which is well-founded.

The well-founded part is the initial part [internally] of $(M,E)$ which is truly well-founded. It is exactly the sets whose [internal] von Neumann rank is an ordinal in the well-founded part of $(Ord^M,E)$.

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    Alright! Thanks! I am perfectly clear about this now.2012-10-26
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This is a definition taken from the proof of Theorem 47 of Azriel Lévy’s monograph, A Hierarchy of Formulas in Set Theory (Memoirs of the AMS, Number 57).

Definition. Let $ M $ be a set, and $ E $ a binary relation on $ M $.

  • A subset $ X $ of $ M $ is called $ E $-transitive if and only if $ (\forall x,y \in M)(((y \in X) \land ((x,y) \in E)) \to (x \in X)). $
  • The $ E $-transitive closure of an element $ x $ of $ M $ is defined as the following subset of $ M $: $ \{ y \in M \mid (\forall X) ( ((x \in X \subseteq M) \land (X ~ \text{is} ~ E \text{-transitive})) \to (y \in X) ) \}. $
  • A subset $ X $ of $ M $ is called $ E $-well-founded if and only if for every non-empty subset $ Y $ of $ X $, there exists a $ y \in Y $ such that $ (x,y) \notin E $ for every $ x \in Y \setminus \{ y \} $.
  • The $ E $-well-founded part of $ M $ is finally defined as the following subset of $ M $: $ \{ x \in M \mid \text{The} ~ E \text{-transitive closure of} ~ x ~ \text{is} ~ E \text{-well-founded} \}. $