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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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    Brian deleted his answer before I could comment, but AC has nothing to do with the von Neumann construction.2012-10-26
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    I must have mistaken it for something else. Which axiom is it that shows that the von Neumann hierarchy equals the whole universe?2012-10-26
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    The fact that every set has a rank is sufficient. This follows from regularity and some replacement (for the transfinite induction defining rank).2012-10-26
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    I mistook it for AC. I now remember that it was Regularity instead. I'm assuming Replacement, of course. Thanks!2012-10-26
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    That has nothing to do with your question, really. Just a side note to what you wrote under Brian's answer.2012-10-26
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    Sure! I understand 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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    Asaf: It is not just the ordinal part. It is the subclass of $M$ consisting of those sets $x\in M$ such that $E$ restricted to $x$ is well-founded. This is a model of KP (though in general, not an inner model of $M$, or even a definable subclass of $M$).2012-10-26
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    @Andres: But the ranks of the well-founded sets are exactly the well-founded ordinals, right?2012-10-26
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    I remember John Steel himself mentioning that the ordinal part is a subset of the well-founded part. Is it possible that we are looking at the external transitive closure of the ordinal part? Just to inform you, I may not be making sense here.2012-10-26
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    To be somewhat more precise, given $x\in M$, we can identify it with $x^*=\{y\in M\mid M\models y E x\}$. When I talk of the transitive closure of $x$, I really mean its $*$ version, so we look at $x$, and $x^*$, and $\{z\in M\mid M\models z E \bigcup x\}$ and $\dots$. If $E$ restricted to this collection is well-founded, then we say that $x$ is in the well-founded part of $M$. (From this it easily follows that yes, the ranks of well-founded sets are the well-founded ordinals of the model).2012-10-26
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    @Andres: I think that you're much more capable of writing an answer. I'll add the important correction, but I'd still love reading an answer by you.2012-10-26
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    @Asaf: What you mean by 'an ordinal in the well-founded part of $ (\text{Ord}^{M},E) $' may not be an actual ordinal in the universe, right? I interpret it to mean that $ (\text{Ord}^{M},E) $ can be Mostowski-collapsed to some initial segment of $ (\text{Ord},\in) $. I apologize for being so fussy about the details.2012-10-26
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    @Haskell: No need to apologize. Recall that $M$ is a set, so $Ord^M$ is also a set. It has a maximal initial segment which is well-founded; look at it as the least [true] ordinal which cannot be embedded into an initial segment of $Ord^M$. It might be everything, in which case the model is well-founded, or it might be just an initial segment of some length. In either way, there are ordinals which lie in this initial segments, and they are the ranks of the well-founded part of $M$.2012-10-26
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    I see... Now, suppose we have an embedding $ \eta $ from some true ordinal into an initial segment of $ (\text{Ord}^{M},E) $. Is $ \eta $ actually the inclusion map up to some point? Equivalently, are the 'internal' ordinals in $ (\text{Ord}^{M},E) $ precisely the same as true ordinals up to a certain point?2012-10-26
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    @Haskell: Of course it does not have to be the inclusion map. However you can collapse the well-founded part of the universe, and then it will be the inclusion map. To see that it doesn't have to be, take a well-founded model $M$ now replace every element $a$ by $a\times\{M\}$, and define $a\times\{M\} E b\times\{M\}$ if and only if $a\in b$. It is well-founded and all, but no map from the ordinals into $M$ is an inclusion map.2012-10-26
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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 $: $$ \{ x \in M \mid \text{The} ~ E \text{-transitive closure of} ~ x ~ \text{is} ~ E \text{-well-founded} \}. $$