Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 1

An inner model of ZFC is a transitive class containing every ordinal and satisfying all its axioms. Write this class as $A$ with its inherited membership relation. Satisfaction is interpreted by restricting all quantifiers to $A$. In a first-order formulation this is a schema for a definable class, possibly with fixed parameters. The class may equal the whole universe. Transitivity means $x\in y\in A\Rightarrow x\in A$; containing all ordinals rules out treating an arbitrary transitive set model as an inner model.

A strongly inaccessible cardinal is an uncountable cardinal $\lambda$ that is both regular and a strong limit:Regularity excludes expressing $\lambda$ as the supremum of a shorter increasing sequence, while the strong-limit requirement concerns all smaller power sets. Merely being an uncountable regular limit cardinal is the weaker notion of a weakly inaccessible cardinal.

A first-order formula $\phi (x_1,\dots ,x_n)$ is an absolute formula for membership structures $A$ and $B$ if, for every tuple $\vec{a}\in (A\cap B{)}^n$,The same elements interpret the free variables in both structures; bound variables range over their respective domains. Thus absoluteness requires both directions, rather than only preservation of truth from the smaller structure to the larger one.

Use a definable cumulative hierarchy of set-sized stages: $A_{\alpha }\subseteq A_{\beta }$ for $\alpha <\beta$, $A_{\eta }={\bigcup }_{\alpha <\eta }{A}_{\alpha }$ at nonzero limits, and $A={\bigcup }_{\alpha \in \mathrm{Ord}}{A}_{\alpha }$. Class parameters and the hierarchy are fixed definable data. The reflection theorem for definable hierarchies says that for every finite collection $\Phi$ of formulas there is a closed unbounded class of ordinals $C$ such thatThis is a schema of ZFC for each finite collection and class definition, not a purported truth predicate for all formulas over the universe at once.

Close $\Phi$ under subformulas. For each existential subformula $\exists y\psi (y,\vec{x})$ and each tuple in $A_{\alpha }$, if a witness exists in $A$, take the least stage index containing a witness. There are only set-many parameter tuples and finitely many formulas. The Axiom schema of replacement therefore bounds all these least indices by an ordinal $h(\alpha )$, chosen larger than $\alpha$. No definable selection of the witnesses themselves is needed.

Above any prescribed bound choose ${\alpha }_0<{\alpha }_1<\cdots$ with ${\alpha }_{m+1}>h({\alpha }_m)$, and put $\eta ={\sup }_m{\alpha }_m$. Every tuple in $A_{\eta }$ lies in some $A_{{\alpha }_m}$, and each true existential instance for that tuple has a witness in $A_{{\alpha }_{m+1}}\subseteq A_{\eta }$. Induction over the subformulas now proves agreement between $A_{\eta }$ and $A$: atomic formulas use the same membership relation, Boolean operations preserve agreement, and the existential step uses this witness property. Thus reflecting stages are unbounded.

For closedness, suppose reflecting stages have limit $\eta$. Every tuple in $A_{\eta }$ lies in a reflecting stage below $\eta$, and every true existential instance has a witness there. The same subformula induction proves reflection at $\eta$. Hence the reflecting stages for the subformula-closed collection form a closed unbounded class, proving the Lévy reflection theorem in this relative form.

The printed inclusion-and-elementarity assertion is false if transitivity is required of the same submodel. Take the theorem of ZFC that combines the axiom of infinity and the Axiom of power set. Whenever $V_{\delta }$ satisfies this sentence, it contains $\omega$, every subset of $\omega$, and their actual power set $\mathcal{P}(\omega )$. A submodel $M\prec V_{\delta }$ contains $\omega$ and $\mathcal{P}(\omega )$, since these are uniquely definable in $V_{\delta }$. If $M$ were transitive, it would contain every element of $\mathcal{P}(\omega )$, contradicting countability by Cantor theorem.

The corrected conclusion uses an elementary embedding rather than elementary inclusion. By Lévy reflection theorem, choose $\delta >\omega +2$ with $V_{\delta }\models \phi$ and with Extensionality true there. The Downward Lowenheim-Skolem theorem says that an infinite structure in a countable language has a countable elementary substructure. Apply it to obtain a countable $N\prec (V_{\delta },\in )$. The membership relation on $N$ is externally well-founded, and elementarity makes it extensional. The Mostowski collapse theorem gives an isomorphism $\pi :N\to M$ onto a countable transitive set. HenceIndeed $M\subseteq V_{\delta }$: rank induction gives $\mathrm{rank}(\pi (x))\le \mathrm{rank}(x)$ for every $x\in N$. The crucial correction is that $j$, rather than the inclusion of $M$, is elementary. For a formula with free variables, apply this argument to its universal closure.

Let $\theta ={\xi }^{\ast }$, and work with the set structure $V_{\theta }\models \mathrm{ZFC}$. Its ordinal height is an infinite limit: a transitive model of ZFC has no largest ordinal, since it can take the successor of each ordinal it contains. Thus $\mathrm{cf}(\theta )\ge \omega$.

Suppose instead that $\mathrm{cf}(\theta )>\omega$. Enumerate all first-order formulas, and close each finite initial collection under subformulas. Applying Lévy reflection theorem inside $V_{\theta }$, choose a strictly increasing sequence ${\beta }_m<\theta$ such that $V_{{\beta }_m}$ agrees with $V_{\theta }$ on the first $m$ collections. Take $\eta ={\sup }_m{\beta }_m<\theta$. The internal rank levels here are the actual rank levels, because $V_{\theta }$ is a transitive rank model.

If $V_{\theta }\models \exists x\psi (x,\vec{a})$ with $\vec{a}\in V_{\eta }$, choose $m$ large enough to include this formula and all the parameters in $V_{{\beta }_m}$. Reflection supplies a witness in $V_{{\beta }_m}\subseteq V_{\eta }$. The Tarski-Vaught test therefore gives $V_{\eta }\prec V_{\theta }$. In particular $V_{\eta }$ satisfies all of ZFC, contrary to the minimality of $\theta$.

ConsequentlyThe countable enumeration is of formulas, not merely axioms: witness closure is what makes the union a model of the entire theory.

The structure $H_{\kappa }$ of hereditarily small sets is transitive. If $\kappa \le \omega$, all its sets are hereditarily finite, so it cannot satisfy Infinity. Therefore the assumed model has $\kappa >\omega$.

Let $\mu <\kappa$ be a cardinal. The ordinal $\mu$ belongs to $H_{\kappa }$, and every actual subset of $\mu$ also belongs to $H_{\kappa }$: its transitive closure has size at most $\max (\mu ,\omega )<\kappa$. The Power set axiom inside $H_{\kappa }$ therefore produces the actual $\mathcal{P}(\mu )$, since all subsets relevant to the internal definition are present. This power set itself belongs to $H_{\kappa }$, soThus $\kappa$ is a strong limit. Its regularity was assumed, and we have proved uncountability. Hence $\kappa$ is strongly inaccessible.