New Foundations is consistent

4.4 Proving freedom of action

Definition 4.19 approximates

We say that a \( \beta \)-approximation \( \psi \) approximates a \( \beta \)-allowable permutation \( \rho \) if \( \psi _A^{\mathcal L}\leq \rho _A^{\mathcal L}\) and \( \psi _A^{\mathcal A}\leq \rho _A^{\mathcal A}\) for each path \( A : \beta \rightsquigarrow \bot \). If \( \psi \) approximates \( \rho \) then \( \psi ^n \) approximates \( \rho ^n \) for each \( n : \mathbb Z \). 1 A \( \beta \)-approximation \( \psi \) exactly approximates a \( \beta \)-allowable permutation \( \rho \) if \( \psi \) approximates \( \rho \), and in addition, if \( a \) is an atom and \( A : \beta \rightsquigarrow \bot \), then \( a \notin \operatorname {\mathsf{coim}}\psi _A^{\mathcal A}\) implies \( \rho (a)^\circ = \rho (a^\circ ) \) and \( \rho ^{-1}(a)^\circ = \rho ^{-1}(a^\circ ) \).

Definition 4.20 freedom of action
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We say that freedom of action holds at a type index \( \delta \) if every coherent \( \delta \)-approximation exactly approximates some \( \delta \)-allowable permutation.

Proposition 4.21 adding flexible litters

Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( L \) be \( A \)-flexible. Then there is a coherent extension \( \chi \geq \psi \) with \( L \in \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\).

Proof

Define \( L' : \mathbb Z \to {\mathcal L}\) by \( L'(n) = L \), then appeal to proposition 4.15 to obtain \( \chi \geq \psi \). All we must do is check that \( \psi \) is coherent at \( (L, L) \), which is trivial.

Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( L \) be \( A \)-inflexible with path \( (\gamma , \delta , \varepsilon , B) \) and tangle \( t : \mathsf{Tang}_\delta \). Suppose that \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). 2 Suppose that freedom of action holds at level \( \delta \). Then there is a coherent extension \( \chi \geq \psi \) with \( L \in \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\).

Proof

Let \( \rho \) be a \( \delta \)-allowable permutation that \( (\psi _B)_\delta \) approximates. Then for each \( n : \mathbb Z \), as \( (\psi ^n_B)_\delta \) approximates \( \rho ^n \), we obtain \( (\psi ^n_B)_\delta (\operatorname {\mathsf{supp}}(t)) = \rho ^n(\operatorname {\mathsf{supp}}(t)) \) as \( (\psi ^n_B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). 3 Define \( L : \mathbb Z \to {\mathcal L}\) by \( L(n) = f_{\delta ,\varepsilon }(\rho ^n(t)) \).

Suppose that there is some \( n \) such that \( L(n) \in \operatorname {\mathsf{coim}}\psi ^{\mathcal L}\). Note that

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

So as \( \psi ^{-n} \) is coherent, we obtain \( (L(n), f_{\delta ,\varepsilon }(t)) \in (\psi ^{-n}_A)^{\mathcal L}\). In particular, \( f_{\delta ,\varepsilon }(t) \in \operatorname {\mathsf{coim}}\psi _A^{\mathcal L}\) already, and no work needs to be done.

We first check the hypothesis of proposition 4.11 for adding orbits. If \( f_{\delta ,\varepsilon }(\rho ^m(t)) = f_{\delta ,\varepsilon }(\rho ^n(t)) \), then \( \rho ^m(t) = \rho ^n(t) \), so \( \rho ^{m+k}(t) = \rho ^{n+k}(t) \), giving \( f_{\delta ,\varepsilon }(\rho ^{m+k}(t)) = f_{\delta ,\varepsilon }(\rho ^{n+k}(t)) \) as required.

We now check the criterion of proposition 4.15 for adding orbits coherently. It suffices to show that \( \psi \) is coherent at \( (L(n), L(n+1)) \) for each \( n : \mathbb Z \). This is witnessed by \( \rho \), which satisfies

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

and

\[ L(n) = f_{\delta ,\varepsilon }(\rho ^n(t));\quad L(n+1) = f_{\delta ,\varepsilon }(\rho (\rho ^n(t))) \]

as required. 4

Proposition 4.23
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If \( (\psi _i)_{i : I} \) is a chain of coherent approximations where \( I \) is a linear order, then the supremum \( \psi \) is coherent.

Proof

Direct, using the same idea as the proof of proposition 4.15.

Freedom of action holds at all type indices \( \beta \leq \alpha \).

Proof

By induction, we may assume freedom of action holds at all \( \delta {\lt} \beta \). Let \( \psi \) be a coherent \( \beta \)-approximation, and let \( \chi \) be a maximal coherent extension, which exists by Zorn’s lemma and proposition 4.23.

Suppose that there is a litter \( L \) such that there exists a path \( A \) where \( L \notin \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\). Let \( L \) have minimal position with this property, and let \( A \) be such a path.

Suppose that \( L \) is \( A \)-flexible. Then by proposition 4.21, there is an extension \( \varphi \) of \( \chi \) such that \( L \in \operatorname {\mathsf{coim}}\varphi _A^{\mathcal L}\), contradicting maximality of \( \chi \).

Suppose that \( L \) is \( A \)-inflexible, with path \( (\gamma ,\delta ,\varepsilon ,B) \) and tangle \( t \). Then \( (\psi _B)_\delta \) is defined on all of \( \operatorname {\mathsf{supp}}(t) \). Indeed, by definition 2.25 (coherent data) and proposition 2.22 (fuzz maps), for each atom or near-litter \( y \) that appears in the range of \( \operatorname {\mathsf{supp}}(t)_C \), we have \( \iota (y) {\lt} \iota (t) {\lt} \iota (L) \), giving the desired conclusion by minimality of the position of \( L \) and the criteria of proposition 2.19. Thus, we obtain the same contradiction by proposition 4.22.

So \( \operatorname {\mathsf{coim}}\chi _A^{\mathcal L}\) is the set of all litters for each path \( A \). We then use the fact that our model data is coherent to recursively compute the allowable permutation \( \rho \) with the same action as \( \chi \). Then \( \chi \) exactly approximates \( \rho \), so \( \psi \) also exactly approximates \( \rho \). 5

  1. We should define what it means for a base approximation to approximate a near-litter permutation, and define this in terms of that.
  2. This is a nontrivial definition to make.
  3. This should of course be its own lemma.
  4. It might be helpful to abstract away the lemma \( (\psi ^m_B)_\delta (\operatorname {\mathsf{supp}}(\rho ^n(t))) = \operatorname {\mathsf{supp}}(\rho ^{n+m}(t)) \) for the two places in the proof where this idea is used.
  5. In general, if \( \psi \leq \chi \) and \( \chi \) (exactly) approximates \( \rho \) then \( \psi \) (exactly) approximates \( \rho \).