13
$\begingroup$

Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.

Are some interesting mathematical questions, which are not resolved in $\mathrm{ZF}$ + $\mathrm{V}=\mathrm{L}$, settled in the Minimal Model of ZF?

$\endgroup$
8
  • 3
    $\begingroup$ en.wikipedia.org/wiki/Minimal_model_(set_theory) $\endgroup$ Apr 21 at 20:48
  • 2
    $\begingroup$ @JoelDavidHamkins It is interesting that all the sets in the minimal model are named by a sentence of $\mathrm{ZF}$. But I do not know that the mentioned naming property may be of help with settling any interesting mathematical questions. $\endgroup$ Apr 21 at 21:06
  • 3
    $\begingroup$ Yes, I agree, being pointwise definable is not expressible in the language of set theory. Nevertheless, I find it an important feature to mention, when one has a model that is pointwise definable. $\endgroup$ Apr 21 at 21:09
  • 2
    $\begingroup$ Yes, it is jstor.org/stable/43303639. $\endgroup$ Apr 21 at 21:16
  • 3
    $\begingroup$ The assertion "there is a definable ..." is not generally expressible, but if the complexity of the definition is known, as it is in the case when $V=L$, then "there is a $\Sigma_2$-definable well-ordering of the reals" is expressible. If $V=L$, we can provide a specific $\Sigma_2$ definition and prove that it is a well-ordering of the entire set-theoretic universe. In fact, the restriction of this order to the reals is projectively definable in the structure of the reals $\langle\mathbb{R},+,\cdot,0,1,<,\mathbb{N}\rangle$, and this is generally the definable order that is meant in your case. $\endgroup$ Apr 22 at 14:42

1 Answer 1

20
$\begingroup$

The minimal model, when it exists, also known as the Shepherdson-Cohen model, is the smallest transitive model of ZFC. This model will have the form $L_\alpha$ for some countable ordinal $\alpha$, and indeed it is $L_\alpha$ where $\alpha$ is the smallest ordinal for which this is a model of ZFC. The minimal model exists whenever there is a well-founded model of ZFC, since in this case the Mostowski collapse of that model will be a transitive model, and the $L$ of that model will be the minimal model or have it as an initial segment.

It is slightly better to refer to the model as the minimal transitive model of ZFC, since it is not actually minimal with respect to being merely a model of ZFC—it will have models inside it that it thinks are models of ZFC. Furthermore, the minimal model is not merely minimal, but least, since it is contained in all other transitive models of ZFC.

Since it is a model of $\text{ZFC}+V=L$, one gets on board all of the consequences of that theory in the minimal model. But there are many further properties true in the minimal model, when it exists, that are not provable from $\text{ZFC}+V=L$.

  • The minimal model satisfies $\text{Con}(\text{ZFC})$ and $\text{Con}(\text{ZFC}+\text{Con}(\text{ZFC}))$ and so forth, iterated many times, since these are arithmetic consequences of the existence of a transtive model of ZFC. These assertions go beyond what is provable in $\text{ZFC}+V=L$ alone, and so they provide examples of interesting assertions provably true in the minimal model but not provable in that theory.

  • Meanwhile, the minimal model satisfies the mildly self-referential assertion "the minimal model of ZFC does not exist". It can't exist inside itself. Consequently, the minimal model thinks there is no transitive model of ZFC. In a sense, the theory of the minimal model of ZFC is exactly on the boundary of the theory of consequences of the existence of a transitive model of ZFC that are absolute to such models, and the assertion that such models actually exist. The minimal model has all those absolute consequences, but not the existence of a transitive model.

  • The minimal model is pointwise definable—every individual is definable without parameters. This is because being a model of $V=L$ it has a definable global well order, and so the definable elements of it will form an elementary substructure, which will Mostowski collapse to some $L_\alpha$, but by minimality the collapse cannot be smaller than minimal model itself, and so it is isomorphic to its definable hull. So everything is definable without parameters.

  • The minimal model thus forms a counterexample ZFC universe for the Math tea argument, which attempts to show that there must be nondefinable real numbers, since there are only countably many definitions in set theory, but uncountably many real numbers.

  • The minimal model has no inaccessible cardinals, nor any worldly cardinals. If it did, then they would provide smaller transitive models, contrary to minimality. In this sense, the minimal model is not great for large cardinal set theory.

  • Similarly, the minimal model can have no inner models of large cardinals, and this will mean, for example, that it must deny certain fragments of determinacy.

  • Meanwhile, if there actually are numerous large cardinals (above the minimal model), then the minimal model, being a transitive model of ZFC, will accurately believe the corresponding consistency assertions, which are absolute to any transitive model, and in this sense, it can still have some access to large cardinal set theory.

  • Indeed, the same idea shows that the minimal model is correct about the consistency of any theory that it can formulate.

  • The consistency strength of the existence of the minimal model is strictly stronger than ZFC, but weaker than the existence of a worldly cardinal, which is itself weaker than the existence of an inaccessible cardinal.

$\endgroup$
6
  • $\begingroup$ For my own edification: This means that, in the minimal model, all alephs exist but not all beths? Or is "all beths exist" independent? $\endgroup$
    – Corbin
    Apr 22 at 19:02
  • $\begingroup$ @Joel David Hamkins "The minimal model satisfies Con(ZFC) and Con(ZFC+Con(ZFC)) and so forth, iterated many times, since these are arithmetic consequences of the existence of a transtive model of ZFC." Transfinitely many times, as Feferman had, for PA, to obtain arithmetical completeness? $\endgroup$ Apr 22 at 19:43
  • 2
    $\begingroup$ @Corbin The minimal model is a model of ZFC, which proves that for every ordinal $\alpha$ both $\aleph_\alpha$ and $\beth_\alpha$ exist. The minimal model is countable, however, so it doesn't have the genuine alephs or beths, but it has ordinals that it thinks fulfill the defining properties of these cardinals. $\endgroup$ Apr 22 at 20:43
  • 2
    $\begingroup$ @FrodeAlfsonBjørdal Yes, you can iterate transfinitely, although this becomes a subtle matter for large ordinals, since one requires an ordinal notation system to express the theories that arise, and this is a necessary aspect in order for the assertions to be expressible. One effect is the phenomenon identified by Feferman that you mention. $\endgroup$ Apr 22 at 20:44
  • 1
    $\begingroup$ Yes, if the minimal model exists, then it will think that ZFC is arithmetically sound. The reason is that if ZFC proves an arithmetic statement $\varphi$ inside the minimal model, then it is really a proof of $\varphi$ in the ambient universe where the minimal model exists, and so $\varphi$ must hold in all ZFC models in that ambient context, and this includes the minimal model. So $\varphi$ is true in that model, as desired. $\endgroup$ Apr 23 at 15:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.