Difference between revisions of "Heights of models"

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(ZFC without the powerset axiom)
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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 strength<sup>Wrong term?</sup> of theories to size of models.
 
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 strength<sup>Wrong term?</sup> 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<!--Taranovsky, "Ordinal Notation", section 6.2-->
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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<!--Taranovsky, "Ordinal Notation", section 6.2-->.
  
 
==KP variants==
 
==KP variants==
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Properties of these models can be couched in terms of elementary substructures: if $L_\alpha\prec L_\beta$ and $\alpha\in\beta$, then $L_\alpha\vDash\textrm{ZFC}^-$. [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf#page=7]
 
Properties of these models can be couched in terms of elementary substructures: if $L_\alpha\prec L_\beta$ and $\alpha\in\beta$, then $L_\alpha\vDash\textrm{ZFC}^-$. [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf#page=7]
  
The latter result extends to:
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The latter result will also hold for these:
 
===ZFC- plus some cardinals===
 
===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 $\alpha_1<\alpha_2$: $\beta_1 < \omega_1^{L_{\beta_2}}$.
 
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 $\alpha_1<\alpha_2$: $\beta_1 < \omega_1^{L_{\beta_2}}$.

Revision as of 22:34, 11 September 2021

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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