New Foundations is consistent

4.3 Structural approximations

Definition 4.12
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For a type index \( \beta \), a \( \beta \)-approximation is a \( \beta \)-tree of base approximations. We define the partial order on \( \beta \)-approximations branchwise. We define an action of \( \beta \)-approximations \( \psi \) on \( \beta \)-supports \( S \) by \( (\psi (S))_A = \psi _A(S_A) \).

Definition 4.13
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Let \( A \) be a \( \beta \)-extended type index. A litter \( L \) is \( A \)-inflexible if there is an inflexible \( \beta \)-path \( I \) such that \( A = ((A_I)_{\varepsilon _I})_\bot \) and \( L = f_{\delta _I, \varepsilon _I}(t) \) for some \( t : \mathsf{Tang}_{\delta _I} \). The coderivative operation works in the obvious way. A litter can be \( A \)-inflexible in at most one way. 1

We say that a \( L \) is \( A \)-flexible if it is not \( A \)-inflexible. 2 If \( L \) is \( B_A \)-flexible, then \( L \) is \( A \)-flexible.

A \( \beta \)-approximation \( \psi \) 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

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

    and

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

    (and hence all \( \delta \)-allowable permutations \( \rho \) such that \( (\psi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \) satisfy \( L_2 = f_{\delta ,\varepsilon }(\rho (t)) \)).

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

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

Proposition 4.15 adding orbits coherently

Suppose that \( \psi \) is an approximation and \( L : \mathbb Z \to {\mathcal L}\) is a function satisfying the hypotheses of proposition 4.11. Let \( \chi \) be the extension produced by the structural version of this result. 3 If \( \psi \) is coherent, and is additionally coherent at \( (L(n), L(n+1)) \) for each integer \( n \), then \( \chi \) is coherent.

Proof

This proof just relies on the fact that if \( (\psi _B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \), then the same holds for every extension \( \chi \) of \( \psi \). 4

Proposition 4.16

If \( \psi \) is coherent, then \( \psi ^{-1} \) is coherent.

Proof

Suppose that \( (L_1, L_2) \in (\psi ^{-1}_A)^{\mathcal L}\), so \( (L_2, L_1) \in \psi _A^{\mathcal L}\). Suppose first that \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). If \( L_2 \) were \( A \)-flexible, then \( L_1 \) would also be \( A \)-flexible by coherence, which is a contradiction. So \( L_2 \) is \( A \)-inflexible with path \( (\gamma ', \delta ', \varepsilon ', B') \) and tangle \( t' : \mathsf{Tang}_{\delta '} \), and there is \( \rho : \mathsf{AllPerm}_{\delta '} \) such that

\[ (\psi _{B'})_{\delta '}(\operatorname {\mathsf{supp}}(t')) = \rho (\operatorname {\mathsf{supp}}(t')) \]

and

\[ A = (B_{\varepsilon '})_\bot ;\quad L_2 = f_{\delta ',\varepsilon '}(t');\quad L_1 = f_{\delta ',\varepsilon '}(\rho (t')) \]

We thus deduce \( \varepsilon = \varepsilon ' \) and \( \gamma = \gamma ' \) by the equations for \( A \). By the equation \( L_1 = f_{\delta ,\varepsilon }(t) \), we also obtain \( \delta = \delta ' \) and \( t = \rho (t') \). Then we find

\begin{align*} (\psi _B)_\delta (\operatorname {\mathsf{supp}}(\rho ^{-1}(t))) & = \rho (\operatorname {\mathsf{supp}}(\rho ^{-1}(t))) \\ (\psi _B)_\delta (\rho ^{-1}(\operatorname {\mathsf{supp}}(t))) & = \rho (\rho ^{-1}(\operatorname {\mathsf{supp}}(t))) \\ (\psi _B)_\delta (\rho ^{-1}(\operatorname {\mathsf{supp}}(t))) & = \operatorname {\mathsf{supp}}(t) \\ \rho ^{-1}(\operatorname {\mathsf{supp}}(t)) & = (\psi ^{-1}_B)_\delta (\operatorname {\mathsf{supp}}(t)) \end{align*}

where the last equation uses the fact that \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t') \). Finally, the equation \( L_2 = f_{\delta ,\varepsilon }(\rho ^{-1}(t)) \) gives coherence at \( (A, L_1, L_2) \) as required.

Now suppose that \( L_1 \) is \( A \)-flexible. If \( L_2 \) were \( A \)-inflexible, then so would be \( L_1 \) by coherence. So \( L_2 \) is \( A \)-flexible, as required.

Proposition 4.17

If \( \psi \) and \( \chi \) are coherent and have equal coimages along all paths, then \( \psi \circ \chi \) is coherent.

Proof

Suppose that \( (L_1, L_3) \in ((\psi \circ \chi )_A)^{\mathcal L}\), so \( (L_1, L_2) \in \psi _A^{\mathcal L}\) and \( (L_2, L_3) \in \chi _A^{\mathcal L}\). Suppose that \( L_1 \) is \( A \)-inflexible with inflexible \( \beta \)-path \( I = (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). Then by coherence of \( \psi \), we have \( \rho \) such that

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

and

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

Then \( L_2 \) is \( A \)-inflexible with path \( I \) and tangle \( \rho (t) \). So by coherence of \( \chi \), we have \( \rho ' \) such that

\[ (\psi _B)_\delta (\operatorname {\mathsf{supp}}(\rho (t))) = \rho '(\operatorname {\mathsf{supp}}(\rho (t))) \]

and

\[ L_3 = f_{\delta ,\varepsilon }(\rho '(\rho (t))) \]

As \( \rho '(\operatorname {\mathsf{supp}}(\rho (t))) = \rho '(\rho (\operatorname {\mathsf{supp}}(t))) \), we obtain the desired coherence result.

Instead, if \( L_1 \) is \( A \)-flexible, then so is \( L_2 \) by coherence of \( \psi \), and so is \( L_3 \) by coherence of \( \chi \).

Proposition 4.18

If \( \psi \) is a coherent \( \beta \)-approximation and \( A \) is a path \( \beta \rightsquigarrow \beta ' \), then \( \psi _A \) is a coherent \( \beta ' \)-approximation.

Proof

Let \( (L_1, L_2) \in (\psi _A)_B^{\mathcal L}\). Suppose that \( L_1 \) is \( B \)-inflexible with path \( (\gamma , \delta , \varepsilon , C) \) and \( t : \mathsf{Tang}_\delta \). Then \( L_1 \) is \( A_B \)-inflexible with path \( (\gamma , \delta , \varepsilon , A_C) \) and the same tangle \( t \). So by coherence of \( \psi \), we obtain a \( \delta \)-allowable \( \rho \) such that

\[ (\psi _{(A_C)})_\delta (\operatorname {\mathsf{supp}}(t)) = \rho (\operatorname {\mathsf{supp}}(t)) \]

and

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

This same \( \rho \) can thus be used to establish coherence of \( \psi _A \) at \( (B, L_1, L_2) \).

Thus, by proposition 4.16, whenever \( L_2 \) is \( B \)-inflexible with path \( I \) and tangle \( t \), \( L_1 \) is also \( B \)-inflexible with path \( I \). So if \( L_1 \) is \( B \)-flexible, so is \( L_2 \), as required.

  1. We should make \( A \)-inflexibility into a subsingleton structure.
  2. This is not data, but a proposition.
  3. We need the extension exactly as produced (as data), not an arbitrary extension satisfying the conclusion of the proposition.
  4. Maybe there’s a better lemma to abstract out this idea for this and proposition 4.23?