# Difference between revisions of "Heights of models"

The smallest heights of transitive models of theories can often be large ordinals. If the theory is first-order, consistent and has an infinite model, the least such height must be countable by the Lowenheim-Skolem theorem, so here we don't discuss uncountable models of theories. If we have a first-order theory $$T$$ that's consistent with $$V=L$$, we may take any model $$M$$ of $$T$$, and due to the consistency with the axiom of constructibility, then $$M\prec L_\beta$$ for some large ordinal $$\beta$$[ citation needed ]. We now apply the condensation lemma to obtain a model with $$M'$$, where $$\textrm{Ord}^{(M')}=\textrm{Ord}^{L_\xi}=\xi$$ for some ordinal $$\xi$$.

The smallest transitive models of theories with larger consistency strength usually have larger heights than the smallest transitive models of weaker (by consistency strength) theories, because every model $M$ of the stronger theory usually contains transitive models of the weaker theory, their height is of course less then $\mathrm{Ord}^M$ (the smallest height countable, i.e. less then $\omega_1^M$) and a transitive model in a transitive model is a transitive model.

## Ordering these ordinals

In some cases, we want to prove $$T\not\vdash\exists\alpha(L_\alpha\vDash T)\!"$$ for a first-order theory $$T$$. Since first-order set theories are sound, from  (expand this).

Also, when comparing the models of first-order theories $T_1$ and $T_2$, we may see if $\forall(\varphi\in T_2)(T_1\vdash\varphi)$, i.e. if $T_1$ proves all of $T_2$'s axioms. Then by the completeness theorem, all models of $T_2$ must satisfy all of $T_1$'s axioms as well. This allows us to translate consistency strengthWrong term? of theories to size of models.

If the theory is first-order and consists of (in Dmytro Taranovsky's words) "arbitrary sufficiently satisfiable axioms", the height of its smallest transitive model must be less than the least stable ordinal.

## KP variants

Heights of models of KP are called admissible. Some extensions of KP, such as KP augmented by the Σ1-separation schema, have larger heights of their models. From the logic above, a model of KP+Σ1-sep. is nonprojectible

## ZFC without the powerset axiom

Heights of models of ZFC without the powerset axiom are given in MarekSrebrny73. One of the main results (theorem 2.7) is that $L_\alpha \models \mathrm{ZFC}^- + V=\mathrm{HC}$ iff $\alpha$ starts a gap, i.e. $\alpha$ is a gap ordinal ($$(L_{\alpha+1} - L_\alpha) \cap \mathcal{P} (\omega) = \varnothing$$), but $\forall_{\beta \in \alpha} (L_\alpha - L_\beta) \cap \mathcal{P} (\omega) \neq \varnothing$. ($\mathrm{HC}$ is the class of hereditarily countable sets.)

Properties of these models can be couched in terms of elementary substructures: if $L_\alpha\prec L_\beta$ and $\alpha\in\beta$, then $L_\alpha\vDash\textrm{ZFC}^-$. [1]

The latter result will also hold for these:

### ZFC- plus some cardinals

In $L$, $L_{\omega_{\alpha+1}}$ are uncountable transitive models of ZFC-P (or $\mathrm{ZFC^-}$, ZFC without the power-set axiom) plus "$\omega_\alpha$ exists", so applying Lowenheim-Skolem downward, there must be also countable transitive models of such theories. This argument works in models of ZFC-P too, so the smallest height $\beta_1$ of a model of ZFC-P plus "$\omega_{\alpha_1}$ exists" must be countable in $L_{\beta_2}$, the minimal model of ZFC-P plus "$\omega_{\alpha_2}$ exists", for $\alpha_1<\alpha_2$: $\beta_1 < \omega_1^{L_{\beta_2}}$.

Also, Barwise has proved that for ordinals $$\alpha$$ and $$\beta$$, $$L_\alpha\vDash\beta\textrm{ is a cardinal} >ω\!"$$ implies $$L_\beta\prec_{\Sigma_1}L_\alpha$$ (see stable for related substructure conditions).

## ZF and ZFC

The height of the minimal model of ZFC is greater than some weakened variants of stability, such as the least (+1)-stable, the least inaccessibly-stable, and the least nonprojectible. However, it's less than the least stable ordinal. [1]

David and Friedman have given a characterization of $\textrm{ZF}$-spectra, and in the $\kappa=\omega$ case we get countable spectra. The characterization includes cardinals $\beta$ where $L_\alpha\vDash\beta\textrm{ is regular}\!"$ [2]

## References

1. Madore, David. A zoo of ordinals. , 2017. www   bibtex
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