Shelah's model separates universal Baire property from universal measurability

By [F8], take a countable first-order model $M\models \mathrm{Z}\mathrm{F}\mathrm{C}$; it may be externally ill-founded. Perform the following construction internally to $M$. Its constructible universe $L^M$ satisfies ZFC+GCH. If $M$ thinks that $L^M$ has no inaccessible, set $N_0=L^M$. Otherwise let $\kappa$ be what $L^M$ regards as its least inaccessible and set $N_0=(V_{\kappa }{)}^{L^M}\mathrm{.}$ The internal instance of [F2] says that this rank segment satisfies ZFC and that every internally inaccessible ordinal below $\kappa$ would already be inaccessible in $L^M$, contrary to the internal minimality of $\kappa$. Since $L^M\models V=L$ and internally every member of this inaccessible rank segment has transitive closure of size below $\kappa$, its constructible rank is below $\kappa$; the internal constructibility recursion [F1] therefore gives $N_0\models V=L$. Thus in both cases $N_0$ is an externally countable first-order model of ZFC+$V=L$+"there is no inaccessible cardinal", and hence of ZFC+CH. This is an internal model construction; no external well-foundedness or transitivity of $M$ or $N_0$ is asserted.

Inside $N_0$, apply the direct ZFC+CH construction theorem in [F3] and let $P$ be the forcing it produces. Externally enumerate all dense subsets of the Boolean completion of $P$ that belong to the countable structure $N_0$, recursively meet them, and let $G$ be the generated $N_0$-generic ultrafilter. Form $N_0[G]$ as the Boolean-valued quotient of the internal $N_0$-names: equality and membership of two quotient classes are determined by whether their internal Boolean values lie in $G$. The internal fixed-formula truth theorem from [F8] validates every standard formula and axiom used here; no external recursion through the possibly ill-founded name relation is required. In $N_0[G]$ form the definable inner class $N=HOD(S)$. Because step 1.1 arranged $N_0\models V=L$, the inner-model conclusions in [F3] and [F4] apply and make $N$ a first-order model of ZF+DC in which every set of reals has the Baire property.

In addition, [F4] says internally in $N_0[G]$ that $N$ is transitive and has all of the extension's ordinals and reals. This is the hypothesis needed for the internal constructibility comparison below.

We first compute $L$ across the forcing extension without invoking the external transitivity clause of [F1]. The standard ZFC proof formalised by the forcing theorem says that set forcing adds no ordinals. It then proves, by internal induction on the common ordinals, that $L_{\alpha }^{N_0[G]}=L_{\alpha }^{N_0}$ for every internal ordinal $\alpha$: the zero and limit steps are immediate, and at a successor both sides take the definable subsets of the same preceding set structure, whose first-order satisfaction relation is unchanged. Since $N_0\models V=L$, the union of the ground levels is all of $N_0$. Therefore $N_0[G]$ internally satisfies $L^{N_0[G]}=N_0\mathrm{.}$ This is a theorem proved and evaluated inside the arbitrary model, not an external absoluteness comparison between transitive universes.

Now reason inside $N_0[G]$. The class $N$ is there a definable transitive ZF inner model containing every ordinal by step 3.1. The internal instance of the ZF theorem [F1] therefore gives $L^N=L^{N_0[G]}=N_0\mathrm{.}$ Consequently $N$ satisfies that its constructible universe has no inaccessible cardinal, because that is exactly the first-order property arranged internally in $N_0$ at step 1.1. This establishes the same-$L$ invariant without ever treating the externally ill-founded structures as transitive.

Suppose toward a contradiction that every set of reals in $N$ is Lebesgue measurable. DC gives Countable Choice by [F7]. Since step 4.1 makes ${\omega }_1^N$ noninaccessible in $L^N$, [F5] supplies a real $x\in N$ with ${\omega }_1^{L[x]}={\omega }_1^N$. For every real $r\in N$, the set $A(x\oplus r)$ is a set of reals in $N$ and is therefore measurable by the supposition. This is the full uniform premise of [F6], not just its instance at $r=0$, so $F(x)$ is rapid. But $F(x)$ is itself a set of reals by [F7] and a rapid filter is not Lebesgue measurable, contradicting the supposition. Hence $N$ contains a nonmeasurable set of reals.

Starting from the countable arbitrary model supplied by consistency, steps 1.1--5.1 construct a first-order model $N$ of $\mathrm{Z}\mathrm{F}+\mathrm{D}\mathrm{C}+\text{all BP}+\mathrm{\neg }\text{all LM}$. Hence $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})\to \mathrm{Con}(\mathrm{Z}\mathrm{F}+\mathrm{D}\mathrm{C}+\text{all BP}+\mathrm{\neg }\text{all LM})\mathrm{.}$ No transitive-model consequence of bare consistency is used.

The steps above establish the relative consistency and the failure of the implication from universal BP to universal LM over ZF+DC; this is the Statement.