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Frequently it is useful do deal with countable transitive models M of ZFC, for example in forcing constructions.

The notion of being an ordinal is absolute for any transitive model, so certainly if ($\alpha$ is an ordinal)M then also $\alpha$ is an ordinal. For the same reason, M will contain successors of every ordinal in it.

On the other hand, if M is countable then M cannot contain every countable ordinal; there must be a least (countable) ordinal not in M.

Can anything be said about this ordinal? Does it have any special significance?

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up vote 6 down vote accepted

The least ordinal not in any transitive model of ZFC can also be described as the supremum of the heights of transitive models of ZFC. It is natural here to consider the class S consisting of all ordinals λ for which there is a transitive model of ZFC of height λ. Thus, the ordinal of your title, when it exists, is simply the supremum of the countable members of S. There are a number of relatively easy observations:

  • If M is a transitive model of ZFC, then so is LM, the constructible universe as constructed inside M, and these two models have the same height. Thus, one could equivalently consider only models of ZFC + V = L.

  • It is relatively consistent with ZFC that S is empty, that is, that there are no transitive models of ZFC. For example, the least element of S is the least α such that Lα is a model of ZFC. This is sometimes called the minimal model of ZFC, though of course it refers to the minimal transitive model. It is contained as a subclass of all other transitive models of ZFC. The minimal model has no transitive models inside it, and so it believes S to be empty.

  • The least element of S (and many subsequent elements) is Δ12 definable in V. This is because one can say: a real codes that ordinal iff it codes a well-ordered relation and there is a model of that order type satisfying that no smaller ordinal is in S (a Σ12 property), also iff every well-founded model of ZFC has ordinal height at least the order type of the ordinal coded by z (a Π12 property).

  • If S has any uncountable elements, then it is unbounded in ω1. The reason is that if Lβ satisfies ZFC and β is uncountable, then we may form increasingly large countable elementary substructures of Lβ, whose Mostowski collapses will give rise to increasingly large countable ordinals in S.

  • In particular, if there are any large cardinals, such as an inaccessible cardinal, then S will have many countable members.

  • If 0# exists, then every cardinal is a member of S. This is because when 0# exists, then every cardinal κ is an L-indiscernible, and so Lκ is a model of ZFC. Thus, under 0#, the class S contains a proper class club, and contains a club in every cardinal.

  • S is not closed. For example, the supremum of the first ω many elements of S cannot be a member of S. The reason is that if αn is the nth element of S, and λ = supn αn, then there would be a definable cofinal ω sequence in Lλ, contrary to the Replacement axiom.

  • S contains members of every infinite cardinality less than its supremum. If β is in S, then we may form elementary substructures of Lβ of any smaller cardinality, and the Mostowski collapses of these structures will give rise to smaller ordinals in S.

  • If β is any particular element of S, the we may chop off the universe at β and consider the model Lβ. Below β, the model Lβ calculates S that same as we do. Thus, if β is a limit point of S, then Lβ will believe that S is a proper class. If β is a successor element of S, then Lβ will believe that S is bounded. Indeed, if β is the αth element of S, then in any case, Lβ believes that there are α many elements of S.

  • If S is bounded, then we may go to a forcing extension V[G] which collapses cardinals, so that the supremum of S is now a countable ordinal. The forcing does not affect whether any Lα satisfies ZFC, and thus does not affect S.

Reading your question again, I see that perhaps you meant to consider a fixed M, rather than letting M vary over all transitive models. In this case, you will want to look at fine-structural properties of this particular ordinal. Of course, it exhibits many closure properties, since any construction from below that can be carried out in ZFC can be carried out inside M, and therefore will not reach up to ht(M).

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I did mean to consider a fixed c.t.m. M and ask about the least ordinal not in this particular model. I guess the wording (of the title at least) may have been ambiguous. Your response was interesting to read, though, and may have been an answer to a better question. If you are so inclined, I would be interested if you would expand on your last paragraph. This ordinal cannot be reached from below by things like union, but it would seem that the powerset for example will be different in M than outside. What do you mean by "fine structural properties?" Again, thanks for your answer! –  Kiochi Feb 25 '10 at 18:19
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