New Foundations is consistent

4.6 Structural actions

For a type index \( \beta \), a \( \beta \)-action is a \( \beta \)-tree of base actions. We define an action of \( \beta \)-actions \( \xi \) on \( \beta \)-supports \( S \) by \( (\xi (S))_A = \xi _A(S_A) \).

Definition 4.32

A \( \beta \)-action \( \xi \) is coherent at \( (A, L_1, L_2) \) if:

  • If \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \), then there is some \( \delta \)-allowable permutation \( \rho \) such that

    \[ (\xi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \]

    and

    \[ L_2 = f_{\delta ,\varepsilon }(\rho (t)) \]

    (and hence again every \( \delta \)-allowable \( \rho \) satisfying the hypothesis also satisfies the conclusion).

  • If \( L_1 \) is \( A \)-flexible, then \( L_2 \) is \( A \)-flexible.

We say that \( \xi \) is coherent if whenever \( (L_1, L_2) \in \xi _A^{\mathcal L}\), \( \xi \) is coherent at \( (A, L_1, L_2) \).

Definition 4.33

Let \( A : \beta \rightsquigarrow \bot \). An \( A \)-flexible approximation of a base action \( \xi \) is a base approximation \( \psi \) such that

  1. \( \xi ^{\mathcal A}\leq \psi ^{E{\mathcal A}} \);

  2. if \( L \in \operatorname {\mathsf{coim}}\psi ^{\mathcal L}\), then \( L \) is \( A \)-flexible;

  3. if \( (N_1, N_2) \in \xi ^{\mathcal N}\) and \( N_1^\circ \) is \( A \)-flexible, then \( (N_1^\circ , N_2^\circ ) \in \psi ^{\mathcal L}\);

  4. if \( (N_1, N_2) \in \xi ^{\mathcal N}\), then \( N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ ) \subseteq \operatorname {\mathsf{coim}}\psi ^{\mathcal A}\) and \( N_2 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_2^\circ ) \subseteq \operatorname {\mathsf{coim}}\psi ^{\mathcal A}\);

  5. if \( (N_1, N_2) \in \xi ^{\mathcal N}\), then for each atom \( a_2 \),

    \[ a_2 \in N_2 \leftrightarrow (\exists a_1 \in N_1,\, (a_1, a_2) \in \psi ^{E{\mathcal A}}) \vee (a_2 \notin \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}} \wedge a_2^\circ = N_2^\circ ) \]

A flexible approximation of a \( \beta \)-action \( \xi \) is a \( \beta \)-approximation \( \psi \) such that for each \( A : \beta \rightsquigarrow \bot \), the base approximation \( \psi _A \) is an \( A \)-flexible approximation of \( \xi _A \). Flexible approximations are coherent.

Every base action has an \( A \)-flexible approximation. Hence, every \( \beta \)-action has a flexible approximation, which can be computed branchwise.

Proof

If \( \xi \leq \zeta \) and \( \psi \) is an \( A \)-flexible approximation for \( \zeta \), then \( \psi \) is an \( A \)-flexible approximation for \( \xi \). So it suffices to prove the result for nice base actions \( \xi \) by proposition 4.30.

Define the permutative relation \( R : {\mathcal L}\to {\mathcal L}\to \mathsf{Prop}\) to be a permutative extension of \( \xi ^{\mathcal L}\), which exists by proposition A.5. Let \( \pi \) be the permutation of litters defined by \( R \), or the identity on any litter not in \( \operatorname {\mathsf{coim}}R \).

Define an orbit restriction \( (t, f, \pi ) \) (definition A.3) for \( \operatorname {\mathsf{field}}\xi ^{\mathcal A}\) by

\[ u = \{ a : {\mathcal A}\mid \forall N \in \operatorname {\mathsf{field}}\xi ^{\mathcal N},\, N^\circ = a^\circ \to a \in N \} ;\quad t = u \setminus \operatorname {\mathsf{field}}\xi ^{\mathcal A} \]

with function \( f : {\mathcal A}\to {\mathcal L}\) defined by \( f(a) = a^\circ \), and litter permutation \( \pi \). We must check that for each litter \( L \), the set \( t \cap \operatorname {\mathsf{LS}}(L) \) has cardinality at least \( \max (\aleph _0, \# \operatorname {\mathsf{field}}\xi ^{\mathcal A}) \). But we can write

\[ t \cap \operatorname {\mathsf{LS}}(L) = \operatorname {\mathsf{LS}}(L) \setminus \left( \operatorname {\mathsf{field}}\xi ^{\mathcal A}\cup \bigcup _{N \in \operatorname {\mathsf{field}}\xi ^{\mathcal N},\, N^\circ = L} (\operatorname {\mathsf{LS}}(L) \setminus N) \right) \]

where the set being removed from \( \operatorname {\mathsf{LS}}(L) \) is small, so \( t \cap \operatorname {\mathsf{LS}}(L) \) is a large set, and \( \aleph _0 \) and \( \# \operatorname {\mathsf{field}}\xi ^{\mathcal A}\) are less than \( \# \kappa \), as required. Then by proposition A.4, there is a permutative relation \( S \geq \xi ^{\mathcal A}\) defined on a small set and contained in \( \operatorname {\mathsf{field}}\xi ^{\mathcal A}\cup t = \operatorname {\mathsf{field}}\xi ^{\mathcal A}\cup u \), such that if

\[ (a_1, a_2) \in S \to (a_1, a_2) \in \xi ^{\mathcal A}\vee \pi (a_1^\circ ) = a_2^\circ \]

Let \( T \) be a permutative extension of the restriction of \( \xi ^{\mathcal L}\) to the \( A \)-flexible litters, with coimage contained entirely in the set of \( A \)-flexible litters, given by proposition A.5. From this, we define a base approximation \( \psi = (S, T) \).

It remains to check that \( \psi \) is an \( A \)-flexible approximation of \( \xi \). Conditions 1–3 are trivial, and condition 4 follows from the fact that we assumed \( \xi \) was nice.

We first show an auxiliary result. Let \( (N_1, N_2) \in \xi ^{\mathcal N}\), and let \( (a_1, a_2) \in S \); we will show that \( a_1 \in N_1 \leftrightarrow a_2 \in N_2 \). Suppose first that \( (a_1, a_2) \in \xi ^{\mathcal A}\), in which case we are done as \( \xi \) is a base action. Instead, we have \( a_1 \notin \operatorname {\mathsf{coim}}\xi ^{\mathcal A}, a_2 \notin \operatorname {\mathsf{im}}\xi ^{\mathcal A}\) and \( \pi (a_1^\circ ) = a_2^\circ \). As \( \xi \) is nice, we must have \( a_1 \in N_1 \leftrightarrow a_1^\circ = N_1^\circ \). Similarly, \( a_2 \in N_2 \leftrightarrow a_2^\circ = N_2^\circ \). So if \( a_1 \in N_1 \), we conclude that \( a_2^\circ = \pi (a_1^\circ ) = \pi (N_1^\circ ) = N_2^\circ \) giving \( a_2 \in N_2 \), and if \( a_2 \in N_2 \), we find \( \pi (a_1^\circ ) = a_2^\circ = N_2^\circ \) so \( a_1^\circ = N_1^\circ \), giving \( a_1 \in N_1 \).

We now prove condition 5, which is the equation

\[ a_2 \in N_2 \leftrightarrow (\exists a_1 \in N_1,\, (a_1, a_2) \in S) \vee (a_2 \notin \operatorname {\mathsf{coim}}S \wedge a_2^\circ = N_2^\circ ) \]

where \( (N_1, N_2) \in \xi ^{\mathcal N}\). Consider the first the case where \( (a_1, a_2) \in S \) and \( a_1 \in N_1 \). The auxiliary lemma shows that \( a_2 \in N_2 \) as required. Now consider the case where \( a_2 \notin \operatorname {\mathsf{coim}}S \) and \( a_2^\circ = N_2^\circ \). If \( a_2 \notin N_2 \), then \( a_2 \in N_2 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_2^\circ ) \subseteq \operatorname {\mathsf{im}}\xi ^{\mathcal A}\), a contradiction. Finally suppose that neither holds, so

\[ (\forall a_1,\, (a_1, a_2) \in S \to a_1 \notin N_1) \wedge (a_2 \in \operatorname {\mathsf{coim}}S \vee a_2^\circ \neq N_2^\circ ) \]

If \( a_2 \in \operatorname {\mathsf{coim}}S \), then there is \( a_1 \) such that \( (a_1, a_2) \in S \), and we have \( a_1 \notin N_1 \), giving \( a_2 \notin N_2 \) by the auxiliary lemma. Finally, if \( a_2 \notin \operatorname {\mathsf{coim}}S \) and \( a_2^\circ \neq N_2^\circ \), then \( a_2 \notin N_2 \), since \( a_2 \in N_2 \) would imply \( a_2 \in N_2 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_2^\circ ) \), contradicting the fact that \( \xi \) is nice.

Definition 4.35 approximates

We say that a \( \beta \)-action \( \xi \) approximates a \( \beta \)-allowable permutation \( \rho \) if \( \xi _A^{\mathcal N}\leq \rho _A^{\mathcal N}\) and \( \xi _A^{\mathcal A}\leq \rho _A^{\mathcal A}\) for each path \( A : \beta \rightsquigarrow \bot \). 1

Let \( \xi \) be a base action, and let \( \psi \) be an \( A \)-flexible approximation of it. Let \( \pi \) be a base permutation that \( \psi \) exactly approximates. If \( (N_1, N_2) \in \xi ^{\mathcal N}\) and \( \pi (N_1^\circ ) = N_2^\circ \), then \( \pi (N_1) = N_2 \).

Proof

First, note that

\begin{align*} \pi [N_1] & = \pi [\operatorname {\mathsf{LS}}(N_1^\circ )] \mathrel {\triangle }\pi [N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ )] \\ & = (\pi [\operatorname {\mathsf{LS}}(N_1^\circ ) \cap \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}] \cup \pi [\operatorname {\mathsf{LS}}(N_1^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}]) \mathrel {\triangle }\pi [N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ )] \end{align*}

As \( \psi \) exactly approximates \( \pi \) and \( \pi (N_1^\circ ) = N_2^\circ \), we have the equation

\[ \pi [\operatorname {\mathsf{LS}}(N_1^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}] = \operatorname {\mathsf{LS}}(N_2^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}} \]

Combining this with the fact that \( \psi ^{E{\mathcal A}} \leq \pi ^{\mathcal A}\), and that \( N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ ) \subseteq \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}} \), we obtain

\begin{align*} \pi [N_1] & = (\operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{\operatorname {\mathsf{LS}}(N_1^\circ ) \cap \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}} \cup (\operatorname {\mathsf{LS}}(N_2^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}})) \mathrel {\triangle }\operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ )} \\ & = (\operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{\operatorname {\mathsf{LS}}(N_1^\circ ) \cap \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}} \mathrel {\triangle }\operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ )}) \cup (\operatorname {\mathsf{LS}}(N_2^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}) \\ & = \operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{(\operatorname {\mathsf{LS}}(N_1^\circ ) \cap \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}) \mathrel {\triangle }(N_1 \mathrel {\triangle }\operatorname {\mathsf{LS}}(N_1^\circ ))} \cup (\operatorname {\mathsf{LS}}(N_2^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}) \\ & = \operatorname {\mathsf{im}}\psi ^{E{\mathcal A}}|_{N_1 \cap \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}} \cup (\operatorname {\mathsf{LS}}(N_2^\circ ) \setminus \operatorname {\mathsf{coim}}\psi ^{E{\mathcal A}}) \end{align*}

which is equal to \( N_2 \) by part of definition 4.33.

Let \( \xi \) be a coherent \( \beta \)-action, and let \( \psi \) be a flexible approximation for it. If \( \psi \) exactly approximates some allowable permutation \( \rho \), then \( \xi \) approximates \( \rho \).

Proof

First, note that \( \xi _A^{\mathcal A}\leq \psi _A^{E{\mathcal A}} \) and \( \psi _A^{\mathcal A}\leq \rho _A^{\mathcal A}\) give the required result for atoms. Now suppose that \( (N_1, N_2) \in \xi _A^{\mathcal N}\); we must show that \( \rho _A(N_1) = N_2 \). We prove this by induction on \( \iota (N_1) \), generalising over all \( A \).

By proposition 4.36, it suffices to show that \( \rho _A(N_1^\circ ) = N_2^\circ \). Suppose that \( N_1^\circ \) is \( A \)-flexible. Then by definition 4.33, \( (N_1^\circ , N_2^\circ ) \in \psi _A^{\mathcal L}\). Hence \( \rho _A(N_1^\circ ) = N_2^\circ \) as required.

Suppose not, so \( N_1^\circ \) is \( A \)-inflexible with path \( (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t : \mathsf{Tang}_\delta \). By coherence of \( \xi \), we know that \( (\xi _B)_\delta \) is defined on \( \operatorname {\mathsf{supp}}(t) \), and it suffices to show that

\[ (\xi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = (\rho _B)_\delta (\operatorname {\mathsf{supp}}(t)) \]

Let \( C : \delta \rightsquigarrow \bot \) and \( a \) be an atom such that \( (i, a) \in \operatorname {\mathsf{supp}}(t)_C^{\mathcal A}\) for some \( i \). Then \( (a, ((\rho _B)_\delta )_C(a)) \in ((\xi _B)_\delta )_C^{\mathcal A}\) by the result for atoms. Now suppose \( N \) is a near-litter such that \( (i, N) \in \operatorname {\mathsf{supp}}(t)_C^{\mathcal N}\). Then

\[ \iota (N) {\lt} \iota (t) {\lt} \iota (f_{\delta ,\varepsilon }(t)) = \iota (N_1^\circ ) \]

So we may apply the inductive hypothesis, giving \( (N, ((\rho _B)_\delta )_C(N)) \in ((\xi _B)_\delta )_C^{\mathcal N}\) as required.

Theorem 4.38 freedom of action for actions

Every coherent action approximates some allowable permutation.

Proof

Let \( \xi \) be a coherent \( \beta \)-action, and let \( \psi \) be a flexible approximation for it, which exists by proposition 4.34. Then apply theorem 4.24 (freedom of action) to \( \psi \) to obtain a \( \beta \)-allowable permutation \( \rho \) that \( \psi \) exactly approximates. Finally, appeal to proposition 4.37 to conclude that \( \xi \) approximates \( \rho \).

  1. Again, we should define what it means for a base action to approximate a near-litter permutation, and define this in terms of that.