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I am little confused with this. In any model of set theory (or is only in ZFC?), we start with an initial set, and then by transfinite recursion we build the universe of sets, which is a proper class. The universe of sets can differ, for instance, we don't know if $V=L$, or we at least can assume it either true or false and get different models? (or is it different theories?). I guess there are many other different universes beyond $V$ and $L$, either by transfinite recursion or by some other (unknown to me) method.

The specific question is: Is the set of ordinals in any of those models the same?. I know that there are properties of ordinals (or cardinals) that are not absolute. For instance, there are no ordinals that are measurable cardinals in $L$, but an ordinal that is a measurable cardinal in some other model can still belong to $L$, only that in $L$ it doesn't have that property. So, finally my question: do all the universe of sets of set theory have the same ordinals on every model? In other words, is the universe of sets of the same "size" on all models, so that the proper class or ordinals is well defined, and its size the same? (if that is not the case, do $V$ and $L$ have the same number of ordinals?

So in the end, what I want to know is if the proper class of all ordinals is well defined and always the same, or if it is dependent of the particular set theory or model we are considering in a specific case.

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up vote 6 down vote accepted

First note that since $\mathsf{ZFC}$ is just a formal first-order theory, it doesn't pick out exactly what form a model's interpretation of the $\in$ symbol takes. So there are models of $\mathsf{ZFC}$ where the $\in$ relation is something other than the real membership relation. Actually, this point is crucial in the development of Boolean-valued models. In such circumstances, what $M$ thinks of as an ordinal will not be what $\mathbf{V}$ thinks of as an ordinal.

But I assume that you are thinking of models where $\in$ is interpreted as the real membership relation. But we do have the following facts.

  • If $\kappa$ is an inaccessible cardinal, then $V_\kappa \models \mathsf{ZFC}$, and $V_\kappa$ clearly does not contain all ordinals.
  • Since the underlying language of $\mathsf{ZFC}$ is countable, by the Löwenheim–Skolem Theorem (if it is consistent) $\mathsf{ZFC}$ has countable models. Taking any countable model $\mathfrak{M} = ( M , E )$ of $\mathsf{ZFC}$ there must be ordinals of $\mathbf{V}$ which are not in $M$. It might be that what $\mathfrak{M}$ thinks of as ordinals are not the ordinals of $\mathbf{V}$, but even if we are "lucky" $\mathfrak{M}$'s version of ordinals is "correct" (which will occur if $M$ is a transitive set and $E$ is the real membership relation, see below) there must be ordinals of $\mathbf{V}$ not appearing in $M$.

In the second point above I mentioned transitive set models of $\mathsf{ZFC}$ (where $\in$ is interpreted as real membership). The existence of models of this kind is an even stronger assumption than the simple consistency of $\mathsf{ZFC}$, then we cannot show from $\mathsf{ZFC}$ (or even $\mathsf{ZFC}+\mathrm{Con}(\mathsf{ZFC})$) that such exist.

But transitive (possibly proper class) models of $\mathsf{ZFC}$ must agree with $\mathbf{V}$ about what ordinals are (if not "how many" there are):

If $M$ is a transitive model of $\mathsf{ZFC}$, and $a \in M$, then $M \models \text{"}a\text{ is an ordinal"}$ iff $a$ really is an ordinal (in $\mathbf{V}$).

This is due to the fact that there is a $\Delta_0$-formula $\phi(x)$ (i.e., a formula in which all quantifiers can be expressed in the form $(\exists x \in y)$ for $(\forall x \in y)$) which in $\mathsf{ZFC}$ defines the property of being an ordinal, and it is known that $\Delta_0$-formulas are absolute for transitive models.

A transitive model $\mathbf{M}$ of $\mathsf{ZFC}$ which contains all the ordinals (from $\mathbf{V}$) is called an inner model.

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Why does Löwenheim–Skolem imply there are countable transitive models? Is it not possible that there are only uncountable transitive models? – Zhen Lin Apr 11 '13 at 8:58
@ZhenLin: It doesn't, but I've been having trouble formulating this correctly. Check back in a few minutes to see if the edit is better. – arjafi Apr 11 '13 at 9:01
@Zhen Lin: If $M$ is an uncountable standard model, consider $N\subseteq M$ a countable elementary submodel generated from L-S. It is a submodel of a standard model so its $\in$ relation is the real $\in$ and it is well-founded. Collapse it and you have a countable transitive model. Huzzah. – Asaf Karagila Apr 11 '13 at 9:20
@julianfernandez: You touch on a sticky point. It is more natural to ask whether $\mathbf{V}$ really contains all sets. This is a hypothesis that can never be proved, but we have yet to come across a set that does not have a complete analogue in $\mathbf{V}$. (One doesn't usually think of real numbers as sets themselves, but one can construct a set — and auxiliary functions and relations — in $\mathbf{V}$ which has all the usual properties of the real line.) In this way is is natural — though perhaps a bit incorrect — to think of $\mathbf{V}$ as the universe of all sets. [cont...] – arjafi Apr 11 '13 at 19:15
[...inued] (Note that in $\mathsf{ZFC}$ the class $\mathbf{V}$ is defined $\{ x : x = x \}$; it certainly seems like nothing larger can exist!) Would a "super-universe" contain more sets/ordinals than $\mathbf{V}$? Perhaps, but such a construction could not be carried out in $\mathsf{ZFC}$ (the class under consideration could not be formalised in $\mathsf{ZFC}$). I even don't believe that the construction can be carried out in any set theory (which is subject to Gödel's Incompleteness Theorems). I don't believe if this question can be mathematically answered. – arjafi Apr 11 '13 at 19:15

The class of ordinals of a model $M$ will always be a proper class from the point of view of $M$. However, the model might miss some of the universe's ordinals. This is most clearly seen in rank-initial segments of the universe: if $\kappa$ is an inaccessible cardinal, say, then $V_\kappa$ is a model of ZFC, but its ordinals are (externally) just the ordinals below $\kappa$.

On the other hand, most of the time we deal with models that have the same ordinals as the universe (which could be interpreted as all the ordinals). This is the case with inner models (like L, HOD and similar things), forcing extensions, ground models etc. The class of ordinals should really be seen as a central anchoring point which doesn't change when we do our favourite constructions, so that we have at least some common point to refer to.

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