Shelah's Baire-Property Model and Inner-Model Lower Bounds — Examples

For the explicit matched-width case, fix a level $n>\mathrm{ht}(t)$ satisfying $\mathrm{\mid }T\cap 2^n\mathrm{\mid }\ge \mathrm{\mid }S\cap 2^n\mathrm{\mid }$; this is an additional hypothesis for the display, not a consequence of perfection. Write $S\cap 2^n=\{s_1,\dots ,s_k\}$ and choose distinct ${\eta }_1,\dots ,{\eta }_k\in T\cap 2^n$.

Let $T^{\mathrm{\prime }\mathrm{\prime }}$ be the set of all nodes of $T$ together with all nodes ${\eta }_i⌢\tau$ for tails $\tau$ satisfying $s_i⌢\tau \in S$, together with their initial segments; that is, replace the prefix $s_i$ by ${\eta }_i$ rather than concatenate the full old word. Prefixes shorter than $n$ already lie in $T$. Then $T^{\mathrm{\prime }\mathrm{\prime }}$ is a tree containing $T$, its recorded initial tree through height $\mathrm{ht}(t)$ is $t$, it is perfect because the nodes of $T$ keep their splitting extensions and each ${\eta }_i$ inherits the splitting of the perfect tree $S$ below $s_i$, and it is nowhere dense because its body is the union of the nowhere-dense set $[T]$ with the finitely many homeomorphic images of the closed nowhere-dense sets $[S]\cap [s_i]$. Hence $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ is a direct extension of $(t,T)$.

For every $x\in [S]$ there is exactly one $i$ with $1\le i\le k$ and $x\in [s_i]$. Let ${\rho }_i$ be the full level-$n$ permutation swapping ${\eta }_i$ with $s_i$ (the identity if they agree) and leaving all other level words and all subsequent tail bits unchanged. Since [F2] fixes an enumeration of every finite-prefix rearrangement, define $m_i$ to be the least $m$ with ${\pi }_m={\rho }_i$. Then ${\rho }_i^{-1}(x)\in [{\eta }_i]\cap [T^{\mathrm{\prime }\mathrm{\prime }}]\subseteq [U_G]$, because the graft is recorded in the witness tree $T^{\mathrm{\prime }\mathrm{\prime }}$ and every witness tree of a condition in the generic filter is contained in the generic tree. Hence $x\in {\pi }_{m_i}([U_G])$, and the condition $(t,T^{\mathrm{\prime }\mathrm{\prime }})$ forces $[S]\subseteq {\bigcup }_{1\le i\le k}{\pi }_{m_i}([U_G]),$ a finite subunion of the countable meagre envelope of the absorption lemma.

Singleton case displayed level by level: for $A=\{0^{\omega }\}$, choose $n>\mathrm{ht}(t)$, a node $\eta \in T\cap 2^n$, and let $Z=\{\sigma \in 2^{<\omega }:(\mathrm{\forall }j)(2j<\mathrm{\mid }\sigma \mathrm{\mid }\Rightarrow \sigma (2j)=0)\}$. Its body contains $0^{\omega }$, has arbitrarily late free odd coordinates and is nowhere dense because a later even coordinate can be set to $1$; graft $Z$ below $\eta$, so $T^{\mathrm{\prime }\mathrm{\prime }}=T\cup \{\eta ⌢\sigma :\sigma \in Z\}$. At every level $m\ge n$ the graft contributes the nodes $\eta ⌢\sigma$ with $\mathrm{\mid }\sigma \mathrm{\mid }=m-n$ (some may already belong to $T$). The body remains nowhere dense by the finite-union argument of step 1.2; no same-level sibling of $\eta$ is required. The full prefix permutation swapping $0^n$ and $\eta$ sends the grafted branch $\eta ⌢0^{\omega }\in [U_G]$ to $0^{\omega }$, so $0^{\omega }\in \pi ([U_G])$, and this single finite substitution is the whole code at this stage.

The general existence assertion is the exact content of [F2]. Under the additional matched-width hypothesis, steps 1.1--1.3 exhibit the finite graft explicitly, and step 2.1 supplies the unconditional singleton instance. No claim is made that perfection alone yields the width comparison or that the generic tree itself contains every old tree.

Let $s_0,\dots ,s_{2^n-1}$ enumerate all binary strings of length $n$ in the canonical order and put $F_i=[s_i]$ for $i<2^n$, padded by empty sets for $i\ge 2^n$. Every real in $L[x]\cap 2^{\omega }$ extends exactly one of the listed strings, so $L[x]\cap 2^{\omega }\subseteq {\bigcup }_i{F}_i$; this is a countable cover of the required kind.

If $u\mathrm{\ne }v$ both lie in one cylinder $[s]$ with $\mathrm{\mid }s\mathrm{\mid }=n$, then $u$ and $v$ agree on all coordinates below $n$. Their first differing coordinate is therefore at least $n$, so the prefix length defined by [F1] satisfies $h(u,v)\ge n+1$, and in particular ${\bigcup }_{i}H(F_i)\subseteq \{k:k\ge n\}=\omega \setminus n$.

Therefore $\omega \setminus n\in F(x)$ by the defining cover criterion, for every $n<\omega$, so $F(x)$ contains the Fréchet filter.

The case $n=0$ is included: the unique length-$0$ string has cylinder $2^{\omega }$, every pair of distinct reals in it has first differing prefix length at least $1$, and the cover is the single set $2^{\omega }$ padded by empty sets, giving $\omega =\omega \setminus 0\in F(x)$.

The steps above exhibit the cofinite tails as members of $F(x)$ through explicit cylinder covers, which is the claim.

Apply [F1] to the constant function $f(n)=0$. It supplies the uniformly assigned set $N_f$ and says directly that $N_f$ is a null $G_{\delta }$.

Since the given $U$ is open, has measure below one, and contains $N_f$, the capture clause of [F1] gives $f(n)\in {\phi }_U(n)$ for all sufficiently large $n$. Because $f(n)=0$ for every $n$, this is exactly $0\in {\phi }_U(n)$ eventually.

Thus [step 1.1] gives the claimed null $G_{\delta }$, and [step 1.2] gives the claimed eventual, rather than pointwise, capture of the constant function.

The claim under refutation is the usual relative-consistency assertion that an inaccessible-cardinal hypothesis is needed to obtain the all-Baire-property model. To refute that requirement it suffices to produce the model relative to ZFC alone. This reading is weaker than, and must not be replaced by, the formal assertion that the target theory's consistency disproves the consistency of ZFC plus an inaccessible.

By The exact equiconsistency of ZFC and the all-Baire-property model, the theory ZF+DC plus "every set of reals has the Baire property" is equiconsistent with ZFC alone: in particular, $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})⟶\mathrm{Con}(\mathrm{Z}\mathrm{F}+\mathrm{D}\mathrm{C}+\text{all BP})\mathrm{.}$ The construction therefore needs no inaccessible-cardinal assumption, which refutes the requirement fixed in step 1.1. Equiconsistency with ZFC by itself does not prove that the target consistency fails to imply the consistency of a stronger theory, and no such claim is used here.

The comparison with measurability is a separate calibration: by Exact equiconsistency of universal measurability and an inaccessible, universal Lebesgue measurability is equiconsistent with ZFC plus an inaccessible cardinal. This fact neither supplies a separating model nor, by itself, proves a strict nonimplication between the two bare consistency statements; no such inference is made here.

The semantic separation is also witnessed: by Shelah's model separates universal Baire property from universal measurability there is, relative to $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})$, a model of ZF+DC in which every set of reals has the Baire property and some set of reals is not Lebesgue measurable. This shows that the two regularity assertions themselves separate; it is not offered as a proof that one formal consistency statement fails to imply another.

Steps 1.2 and 1.3 refute the alleged need to assume an inaccessible in the relative-consistency construction and identify the established equiconsistency calibrations; step 1.4 supplies the semantic contrast. No lower bound for measurability transfers to the Baire property, and no unproved nonimplication between bare consistency statements is asserted.

The amalgam data are those of [F3]: $O=P_1{\ast }_{P_0}{P}_2$ consists of the pairs admitted by a common positive $B_0$-condition and is ordered coordinatewise. Its canonical dense subset is $D=\{(q_1,q_2)\in O:q_{\mathrm{\ell }}\in D_{\mathrm{\ell }}\text{ for }\mathrm{\ell }=1,2\}\mathrm{.}$ The denseness assertion already includes the synchronization of the two quotient witnesses; it is not inferred from coordinatewise denseness alone.

For $x=(q_1,q_2)\in D$, let $m(x)$ be the least $m$ such that every pair $(q_1^{\mathrm{\prime }},q_2^{\mathrm{\prime }})$ with $q_{\mathrm{\ell }}^{\mathrm{\prime }}<msubsup><mi>E</mi><mi>m</mi><mi\; mathvariant="normal">\ell </mi></msubsup>q_{\mathrm{\ell }}$ is admitted. Existence is the double application of [F2] in the proof of [F3]: a countable directed cover of $P_0$ is used first for $P_1$ and then, after retaining the dense subfamily below the admission witness, for $P_2$. Two reductions in the same directed piece have a common strengthening and hence admit the perturbed pair. The least number $m(x)$ depends only on the two equivalence classes and admission, not on a chosen witness.

No partial-isomorphism extension theorem is needed. The theorem [F3] applies directly to the two named complete embeddings of the arbitrary common complete subalgebra $B_0$. The weak-coordinate maps give complete canonical copies of both factors in the full amalgam, independently of whether those canonical conditions belong to the selected dense presentation $D$.

If $x^{\mathrm{\prime }}=(q_1^{\mathrm{\prime }},q_2^{\mathrm{\prime }})$ has in both coordinates, then the relevant factor classes are unchanged and minimality gives $m(x^{\mathrm{\prime }})=m(x)$. Hence [F3] defines These relations refine with $n$, have countably many classes, and every class is downward directed. In particular every two members of one $E_n$-class are compatible. The diagonal and transfer assertions are the coordinatewise sweetness clauses combined with the same common-admission property; they are not consequences of pairwise compatibility alone.

The canonical embeddings are complete by the exact conclusion of [F3]; no countable-generation hypothesis on $B_0$ is present in that theorem.

Countable chain condition: the amalgam is sweet by [F3] and step 2.1, and a sweet forcing is a countable union of directed sets, hence ccc by [F4].

The steps above exhibit the intrinsic least modulus, the $E_n$-classes, the two canonical complete embeddings and the ccc conclusion for the amalgam over an arbitrary common complete subalgebra, verifying the claimed instance.