CANTORIAN MODELS OF PREDICATIVE $\mathrm {NF}$

Abstract

Tangled Type Theory was introduced by Randall Holmes in [3] as a new way of approaching the consistency problem for $\mathrm {NF}$. Although the task of finding models for this theory is far from trivial (considering it is equiconsistent with $\mathrm {NF}$), ways of constructing models for certain fragments of it have been discovered. In this article, we present a simpler way of constructing models of predicative Tangled Type Theory and consequently of predicative $\mathrm {NF}$. In these new models of predicative $\mathrm {NF}$, the universe is well-orderable and equinumerous to the set of singletons.

1 Introduction

1.1 Tangled type theory

The language ${\mathcal {L}_{\mathrm {TST}}}$ of Simple Type Theory is the many-sorted language of set theory with one binary relation symbol $\varepsilon $ and countably many types (or sorts) indexed by $\mathbb {N}$ . Each variable of ${\mathcal {L}_{\mathrm {TST}}}$ is assigned a unique type, which we indicate by a superscript. The ${\mathcal {L}_{\mathrm {TST}}}$ -formulas are built inductively from the atomic formulas $x^i \varepsilon y^{i+1}$ and $x^i = y^i$ in the usual way.

Simple Type Theory ( $\mathrm {TST}$ ) is axiomatized by two sets of axioms. The Axiom of Extensionality ( $\mathrm {Ext}$ ) is the set of all the following sentences for each type $i \in \mathbb {N}$ :

(Extⁱ⁺¹) $$\begin{align} \forall x^{i+1}, y^{i+1} (x^{i+1} = y^{i+1} \leftrightarrow \forall z^i (z^i \varepsilon x^{i+1} \leftrightarrow z^i \varepsilon y^{i+1})). \end{align} $$

The Axiom of Comprehension ( $\mathrm {Co}$ ) is the set of all the following sentences for each type $i \in \mathbb {N}$ and formula $\phi $ of ${\mathcal {L}_{\mathrm {TST}}}$ :

(Coⁱ⁺¹) $$\begin{align} \forall \bar{u} \exists y^{i+1} \forall x^i (x^i \varepsilon y^{i+1} \leftrightarrow \phi(x^i, \bar{u})), \end{align} $$

where $y^{i+1}$ is not free in $\phi $ . We define $\mathrm {TST} = \mathrm {Ext} + \mathrm {Co}$ . For each $i \in \mathbb {N}$ , we let $\mathrm {CoP}^{i+1}$ be the axiom we get from $\mathrm {Co}^{i+1}$ if we restrict the types of the bound variables in $\phi $ to not exceed i and the types of the free variables in $\phi $ to not exceed $i+1$ . The Axiom of predicative Comprehension ( $\mathrm {CoP}$ ) is the set of all $\mathrm {CoP}^{i+1}$ for $i \in \mathbb {N}$ . We define predicative Simple Type Theory ( $\mathrm {TSTP}$ ) as $\mathrm {TSTP} = \mathrm {Ext} + \mathrm {CoP}$ .

Now, the language ${\mathcal {L}_{\mathrm {TTT}}}$ of Tangled Type Theory is the same as ${\mathcal {L}_{\mathrm {TST}}}$ , but its formulas are built inductively from the atomic formulas $x^i = y^i$ and $x^i \varepsilon y^j$ for $i < j$ . For each function $s \colon \mathbb {N} \to \mathbb {N}$ and each ${\mathcal {L}_{\mathrm {TST}}}$ -formula $\phi $ , we denote by $\phi ^s$ the ${\mathcal {L}_{\mathrm {TTT}}}$ -formula that we get if we replace each type index i of a variable in $\phi $ with $s(i)$ . For each set of ${\mathcal {L}_{\mathrm {TST}}}$ -sentences T, we let

$$\begin{align*}T^\circ = \{ \sigma^s : \sigma \in T \text{ and } s \colon \mathbb{N} \to \mathbb{N} \text{ strictly increasing} \}. \end{align*}$$

Tangled Type Theory ( $\mathrm {TTT}$ ) is defined as $\mathrm {TTT} = \mathrm {TST}^\circ $ , whereas predicative Tangled Type Theory ( $\mathrm {TTTP}$ ) as $\mathrm {TTTP} = \mathrm {TSTP}^\circ $ .

A structure $\mathscr {A}$ for the language ${\mathcal {L}_{\mathrm {TST}}}$ is a sequence $(A_0, A_1, \ldots , \{\varepsilon ^{\mathscr {A}}_{i,i+1}\}_{i \in \mathbb {N}})$ , where $A_0, A_1, \ldots $ are non-empty sets interpreting the countably many types of ${\mathcal {L}_{\mathrm {TST}}}$ , and each $\varepsilon ^{\mathscr {A}}_{i,i+1} \subseteq A_i \times A_{i+1}$ is a binary relation interpreting $\varepsilon $ for type $i \in \mathbb {N}$ . Similarly, a structure $\mathscr {A}$ for the language ${\mathcal {L}_{\mathrm {TTT}}}$ is a sequence $(A_0, A_1, \ldots , \{\varepsilon ^{\mathscr {A}}_{i,j})\}_{i < j}$ , where $A_0, A_1, \ldots $ are non-empty sets, and $\varepsilon ^{\mathscr {A}}_{i,j} \subseteq A_i \times A_j$ , for all $i, j \in \mathbb {N}$ such that $i < j$ .

Let us now introduce the notion of standard transitive ${\mathcal {L}_{\mathrm {TTT}}}$ -structure. First, we fix some notation about tuples in our metatheory. For all n-tuples $u = (u_1, \ldots , u_n)$ and $0 \leq i \leq n$ , where $n> 0$ , we let

$$\begin{align*}(u)_i = \begin{cases} u, & \text{if } i = 0, \\ u_i, & \text{o.w.} \\ \end{cases} \end{align*}$$

Definition 1.1. An ${\mathcal {L}_{\mathrm {TTT}}}$ -structure $\mathscr {A} = (A_0, A_1, \ldots , \{\varepsilon ^{\mathscr {A}}_{i,j}\}_{i < j})$ is standard transitive if:

1. (i) for all $n \in \mathbb {N}$ ,

   $$\begin{align*}A_{n+1} \subseteq \prod^n_{i = 0} {\mathscr{P}}(\{ (u)_i : u \in A_i \}), \end{align*}$$

2. (ii) for all $i < j$ , $u \in A_i$ , and $v \in A_j$ ,

   $$\begin{align*}u \varepsilon^{\mathscr{A}}_{i,j} v \Leftrightarrow (u)_i \in (v)_{i+1}. \end{align*}$$

To simplify notation, we will denote $\mathscr {A}$ as $(A_0, A_1, \ldots , \in )$ .

Note. Let us make the definition above a bit less confusing. First of all, notice that in Tangled Type Theory, every set of type n has n extensions (one for each type below n). The elements of $A_n$ are basically n-tuples that code this fact. More precisely, it follows by the definition of $\varepsilon ^{\mathscr {A}}_{i,n}$ that the extension of a set $v \in A_n$ over type i is its ( $i+1$ )-th projection $(v)_{i+1}$ . It is important to note that the extension of v over type i is a set of i-th projections of elements of $A_i$ and not a set of elements of $A_i$ . So, in the sense we just described, an element of $A_n$ is a tuple of its n extensions over types $0, \ldots , n-1$ . Keep in mind that there is nothing mysterious about the tuples $v = ((v)_1, \ldots , (v)_n)$ in $A_n$ ; each $(v)_i$ is simply an element of ${\mathscr {P}}^i(A_0)$ . It is also worth noting that we imposed no restrictions on $A_0$ , which means that $A_0$ can be any set.

It is always easier to work with standard transitive structures, and as we show below we may always assume that extensional ${\mathcal {L}_{\mathrm {TTT}}}$ -structures (i.e., structures that satisfy $\mathrm {Ext}^\circ $ ) are standard transitive.

Definition 1.2. Let $\mathscr {A}$ , $\mathscr {B}$ be two ${\mathcal {L}_{\mathrm {TTT}}}$ -structures. We say that f is an ${\mathcal {L}_{\mathrm {TTT}}}$ -isomorphism from $\mathscr {A}$ to $\mathscr {B}$ , if f is a sequence $(f_0, f_1, \ldots )$ of functions such that:

1. (i) for all $i \in \mathbb {N}$ , $f_i \colon A_i \to B_i$ is a bijection,

2. (ii) for all $i, j \in \mathbb {N}$ such that $i < j$ , and for all $u \in A_i$ and $v \in A_j$ ,

   $$\begin{align*}u \varepsilon^{\mathscr{A}}_{i,j} v \Leftrightarrow f_i(u) \varepsilon^{\mathscr{B}}_{i,j} f_j(v). \end{align*}$$

When such an ${\mathcal {L}_{\mathrm {TTT}}}$ -isomorphism exists, we say that $\mathscr {A}$ and $\mathscr {B}$ are ${\mathcal {L}_{\mathrm {TTT}}}$ -isomorphic (or just isomorphic).

The next proposition follows easily by induction on the complexity of $\phi $ .

Proposition 1.3. Let $\mathscr {A}$ and $\mathscr {B}$ be two ${\mathcal {L}_{\mathrm {TTT}}}$ -structures. If $f \colon \mathscr {A} \to \mathscr {B}$ is an ${\mathcal {L}_{\mathrm {TTT}}}$ -isomorphism, then for every formula $\phi (x^{i_1}_1, \ldots , x^{i_n}_n)$ of ${\mathcal {L}_{\mathrm {TTT}}}$ and $a_1 \in A_{i_1}, \ldots , a_n \in A_{i_n}$ , we have that

$$\begin{align*}\mathscr{A} \models \phi(a_1, \ldots, a_n) \Leftrightarrow \mathscr{B} \models \phi(f_{i_1}(a_1), \ldots, f_{i_n}(a_n)). \end{align*}$$

Lemma 1.4. Every extensional ${\mathcal {L}_{\mathrm {TTT}}}$ -structure is isomorphic to a standard transitive ${\mathcal {L}_{\mathrm {TTT}}}$ -structure.

Proof Let $\mathscr {A} = (A_0, A_1, \ldots , \{\varepsilon ^{\mathscr {A}}_{i,j}\}_{i < j})$ be an extensional ${\mathcal {L}_{\mathrm {TTT}}}$ -structure. We define a standard transitive ${\mathcal {L}_{\mathrm {TTT}}}$ -structure $\mathscr {B} = (B_0, B_1, \ldots , \in )$ . Let $B_0 = A_0$ and $f_0$ be the identity function on $A_0$ . For $n \in \mathbb {N}$ , we define $B_{n+1} = \operatorname {\mathrm {ran}}(f_{n+1})$ , where $f_{n+1}$ is defined such that for all $u \in A_{n+1}$ ,

$$\begin{align*}f_{n+1}(u) = (\{(f_0(v))_0 : v \varepsilon^{\mathscr{A}}_{0,n+1} u\}, \ldots, \{(f_n(v))_n : v \varepsilon^{\mathscr{A}}_{n,n+1} u\}). \end{align*}$$

It is easy to verify that $\mathscr {B}$ is standard transitive and that f is an ${\mathcal {L}_{\mathrm {TTT}}}$ -isomorphism from $\mathscr {A}$ to $\mathscr {B}$ .

The following lemma establishes a practical criterion for extensionality.

Lemma 1.5. A standard transitive ${\mathcal {L}_{\mathrm {TTT}}}$ -structure $\mathscr {A} = (A_0, A_1, \ldots , \in )$ is extensional iff for all $0 \leq i < n$ , and $u, v \in A_n$ ,

$$\begin{align*}(u)_{i+1} = (v)_{i+1} \Rightarrow u = v. \end{align*}$$

Proof Just notice that $\mathscr {A}$ is extensional iff for all $0 \leq i < n$ ,

$$ \begin{align*} &\mathscr{A} \models \forall u^n, v^n (u^n = v^n \leftrightarrow \forall w^i (w^i \varepsilon u^n \leftrightarrow w^i \varepsilon v^n)) \\ &\Leftrightarrow \forall u, v \in A_n (u = v \leftrightarrow \forall w \in A_i ((w)_i \in (u)_{i+1} \leftrightarrow (w)_i \in (v)_{i+1})) \\ &\Leftrightarrow \forall u, v \in A_n (u = v \leftrightarrow (u)_{i+1} = (v)_{i+1}), \end{align*} $$

where the second equivalence holds because $(u)_{i+1}, (v)_{i+1} \in {\mathscr {P}}(\{ (w)_i : w \in A_i \})$ .

Note. Notice that by the previous lemma, if $\mathscr {A} = (A_0, A_1, \ldots , \in )$ is extensional, then for all $0 \leq i \leq n$ , and $u, v \in A_n$ ,

$$\begin{align*}(u)_{i+1} = (v)_{i+1} \Leftrightarrow u = v. \end{align*}$$

1.2 New foundations

The language ${\mathcal {L}_{\mathrm {NF}}}$ of New Foundations is the usual one-sorted language of set theory, $\{\epsilon \}$ , where $\epsilon $ is a binary relation symbol. New Foundations ( $\mathrm {NF}$ ) is axiomatized by the axioms of $\mathrm {TST}$ if we erase all type superscripts. Similarly, by erasing all type superscripts from the axioms of $\mathrm {TSTP}$ , we get the axioms of predicative $\mathrm {NF}$ ( $\mathrm {NFP}$ ).

What do we know about $\mathrm {NFP}$ ? First of all, we know that it is a rather weak subtheory of $\mathrm {NF}$ since for example its consistency can be proved in $\mathrm {PA}$ (see [1]). We also know that it is finitely axiomatizable (see [4]), and that if the Union axiom

(Union) $$ \begin{align} \forall z \exists y \forall x (x \epsilon y \leftrightarrow \exists v (v \epsilon z \wedge x \epsilon v)) \end{align} $$

is added to it, we get full $\mathrm {NF}$ , i.e., $\mathrm {NFP} + \mathrm {Union} = \mathrm {NF}$ (see [1]). The most interesting fact about $\mathrm {NFP}$ though is that it is consistent with properties (like choice principles; see [1, 3]) that fail in $\mathrm {NF}$ (see [6] or [2]). Below, we describe a way of constructing such models of $\mathrm {NFP}$ .

2 Models of predicative $\mathrm {NF}$

We work in $\mathrm {ZF} + (V = L)$ . We are going to construct a standard transitive model $\mathscr {A}$ of $\mathrm {TTTP}$ . Let $\kappa $ be an infinite cardinal. For all $n \in \mathbb {N}$ , we define recursively a set $X_n$ , an ordinal $\kappa \leq \alpha _n < \kappa ^+$ , and a bijection $f_n \colon \kappa \to X_n$ such that $f_n \in L_{\alpha _{n+1}}$ . Let $\alpha _0 = \kappa $ , $X_0 = L_{\alpha _0}$ , and let $f_0 \colon \kappa \to X_0$ be a bijection. For $n> 0$ , let

$$\begin{align*}X_n = L_{\alpha_n} \cap {\mathscr{P}}(X_{n-1}), \end{align*}$$

where $\alpha _n$ is the least limit ordinal that is greater than $\alpha _{n-1}$ and for which $f_{n-1} \in L_{\alpha _n}$ . We know that $\alpha _n < \kappa ^+$ and $|X_n| = \kappa $ , so there exists some bijection $f_n \colon \kappa \to X_n$ .

Let $A_0 = X_0 $ and for all $n> 0$ , let

$$\begin{align*}A_n = \{ (f_1(\alpha), \ldots, f_n(\alpha)) : \alpha < \kappa\}. \end{align*}$$

We define $\mathscr {A} = (A_0, A_1, \ldots , \{\varepsilon ^{\mathscr {A}}_{i,j}\}_{i < j})$ , where for all $i < j$ , $u \in A_i$ , and $v \in A_j$ ,

$$\begin{align*}u \varepsilon^{\mathscr{A}}_{i,j} v \Leftrightarrow (u)_i \in (v)_{i+1}. \end{align*}$$

Notice that for all $n \in \mathbb {N}$ , since $f_0, \ldots , f_n \in L_{\alpha _{n+1}}$ and $\alpha _{n+1}$ is a limit ordinal, we have

$$\begin{align*}X_0, \ldots, X_n, A_0, \ldots, A_n \in L_{\alpha_{n+1}}. \end{align*}$$

Lemma 2.1. $\mathscr {A}$ is a standard transitive model of   $\mathrm {TTTP}$ .

Proof We have that for all $i \in \mathbb {N}$ ,

$$\begin{align*}\{ (u)_i : u \in A_i \} = \operatorname{\mathrm{ran}}(f_i) = X_i. \end{align*}$$

Therefore, for all $n \in \mathbb {N}$ ,

$$\begin{align*}A_{n+1} \subseteq \prod^n_{i = 0} \operatorname{\mathrm{ran}}(f_{i+1}) = \prod^n_{i = 0} X_{i+1} \subseteq \prod^n_{i = 0} {\mathscr{P}}(X_i) = \prod^n_{i = 0} {\mathscr{P}}(\{ (u)_i : u \in A_i \}), \end{align*}$$

i.e., $\mathscr {A}$ is standard transitive.

We show that $\mathscr {A}$ is an extensional structure. Let $0 \leq i < n$ , and $u, v \in A_n$ . We know that $u = (f_1(\alpha ), \ldots , f_n(\alpha ))$ and $v = (f_1(\beta ), \ldots , f_n(\beta ))$ for some $\alpha , \beta < \kappa $ . If $(u)_{i+1} = (v)_{i+1}$ , then $f_{i+1}(\alpha ) = f_{i+1}(\beta )$ , so since $f_{i+1}$ is 1–1, we have that $\alpha = \beta $ , i.e., $u = v$ . Therefore, by Lemma 1.5, $\mathscr {A}$ is extensional.

It remains to show that $\mathscr {A} \models \mathrm {CoP}^\circ $ . Let $s \colon \mathbb {N} \to \mathbb {N}$ be strictly increasing, and let $\phi (x^i, u^{i_1}_1, \ldots , u^{i_n}_n)$ be some ${\mathcal {L}_{\mathrm {TST}}}$ -formula

$$\begin{align*}\operatorname{\mathrm{Q}}_1 x^{j_1}_1 \ldots \operatorname{\mathrm{Q}}_m x^{j_m}_m \psi(x^{j_1}_1, \ldots, x^{j_m}_m, x^i, u^{i_1}_1, \ldots, u^{i_n}_n), \end{align*}$$

where $\operatorname {\mathrm {Q}}_1, \ldots , \operatorname {\mathrm {Q}}_n$ are quantifiers, $\psi $ is quantifier-free, $\max \{j_1, \ldots , j_m\} \leq i$ , and $\max \{i_1, \ldots , i_n\} \leq i+1$ . We show that $\mathscr {A} \models (\forall u^{i_1}_1, \ldots , u^{i_n}_n \exists y^{i+1} \forall x^{i} (x^{i} \varepsilon y^{i+1} \leftrightarrow \phi (x^{i}, u^{i_1}_1, \ldots , u^{i_n}_n)))^s$ , i.e., that

$$\begin{align*}\mathscr{A} \models \forall u^{s(i_1)}_1, \ldots, u^{s(i_n)}_n \exists y^{s(i+1)} \forall x^{s(i)} (x^{s(i)} \varepsilon y^{s(i+1)} \leftrightarrow \phi(x^{s(i)}, u^{s(i_1)}_1, \ldots, u^{s(i_n)}_n)). \end{align*}$$

Let $u_1 \in A_{s(i_1)}, \ldots , u_n \in A_{s(i_n)}$ . Let

$$ \begin{align*} &y' = \{ (x)_{s(i)} : x \in A_{s(i)} \wedge \operatorname{\mathrm{Q}}_1 x_1 \in A_{s(j_1)} \ldots \operatorname{\mathrm{Q}}_m x_m \in A_{s(j_m)} \\ &\mathscr{A} \models \psi(x_1, \ldots, x_m, x, u_1, \ldots, u_n) \}. \end{align*} $$

Now, by Remark 1.1 and the way $\varepsilon $ is interpreted in standard transitive structures, it follows that the statement $\mathscr {A} \models \psi (x_1, \ldots , x_m, x, u_1, \ldots , u_n)$ is equivalent to a quantifier-free ${\mathcal {L}_{\mathrm {ZF}}}$ -formula that (apart from $x_1, \ldots , x_m, x$ ) has as parameters only the $s(i)+1$ first coordinates of $u_1, \ldots , u_n$ , where each such coordinate is in $L_{\alpha _{s(i)+1}}$ . Moreover, we know that $A_{s(i)}, A_{s(j_1)}, \ldots , A_{s(j_n)} \in L_{\alpha _{s(i)+1}}$ because $s(j_1), \ldots , s(j_n) \leq s(i)$ . Therefore, since $\alpha _{s(i)+1}$ is a limit ordinal, we have that

$$\begin{align*}y' \in L_{\alpha_{s(i)+1}} \cap {\mathscr{P}}(X_{s(i)}) = X_{s(i)+1}. \end{align*}$$

Let $y \in A_{s(i+1)}$ such that $(y)_{s(i)+1} = y'$ . Clearly, y witnesses that

$$\begin{align*}\mathscr{A} \models \exists y^{s(i+1)} \forall x^{s(i)} (x^{s(i)} \varepsilon y^{s(i+1)} \leftrightarrow \phi(x^{s(i)}, u_1, \ldots, u_n)).\\[-34pt] \end{align*}$$

We now show that $\mathscr {A}$ inherits some interesting properties from L. Below, we present two such properties. We begin by proving that in $\mathscr {A}$ every universe is well-orderable. Before we proceed though, let us examine what it means for an element to be a Wiener–Kuratowski pair in $\mathscr {A}$ . For each $i \in \mathbb {N}$ , let $\mathrm {Pair}_i(u^i, v^i, z^{i+2})$ be the following ${\mathcal {L}_{\mathrm {TST}}}$ -formula:

$$ \begin{align*} &\exists x^{i+1}_1, x^{i+1}_2 (\forall w^i (w^i \varepsilon x^{i+1}_1 \leftrightarrow w^i = u^i) \wedge \forall w^i (w^i \varepsilon x^{i+1}_2 \leftrightarrow w^i = u^i \vee w^i = v^i) \\ & \quad \wedge \forall x^{i+1} (x^{i+1} \varepsilon z^{i+2} \leftrightarrow x^{i+1} = x^{i+1}_1 \vee x^{i+1} = x^{i+1}_2)), \end{align*} $$

expressing that $z^{i+2}$ is the Wiener–Kuratowski pair of $u^i, v^i$ . Notice that for $s \colon \mathbb {N} \to \mathbb {N}$ strictly increasing, $i \in \mathbb {N}$ , $u, v \in A_{s(i)}$ , and $z \in A_{s(i+2)}$ , we have that $\mathscr {A} \models \mathrm {Pair}^s_i(u, v, z)$ is equivalent to

$$ \begin{align*} &\exists x_1, x_2 \in A_{s(i+1)} (\forall w \in A_{s(i)} ((w)_{s(i)} \in (x_1)_{s(i)+1} \leftrightarrow (w)_{s(i)} = (u)_{s(i)}) \\ &\quad\wedge (\forall w \in A_{s(i)} ((w)_{s(i)} \in (x_2)_{s(i)+1} \leftrightarrow (w)_{s(i)} = (u)_{s(i)} \vee (w)_{s(i)} = (v)_{s(i)}) \\ &\quad\wedge \forall x \in A_{s(i+1)} ((x)_{s(i+1)} \in (z)_{s(i+1)+1} \\ &\quad\leftrightarrow (x)_{s(i+1)} = (x_1)_{s(i+1)} \vee (x)_{s(i+1)} = (x_2)_{s(i+1)})), \end{align*} $$

or in more compact notation iff

$$ \begin{align*} &\exists x_1, x_2 \in A_{s(i+1)} ( (z)_{s(i+1)+1} = \{ (x_1)_{s(i+1)}, (x_2)_{s(i+1)} \} \\ &\quad\wedge (x_1)_{s(i)+1} = \{(u)_{s(i)}\} \wedge (x_2)_{s(i)+1} = \{(u)_{s(i)}, (v)_{s(i)}\}). \end{align*} $$

To simplify notation, let us denote by $\mathrm {Pair}^{\mathscr {A},s}_i(u, v, z)$ the above ${\mathcal {L}_{\mathrm {ZF}}}$ -formula which is equivalent to $\mathscr {A} \models (\mathrm {Pair}_i(u, v, z))^s$ .

Lemma 2.2. $\mathscr {A} \models (\{\text {"}V^{i+1} \text { is well-orderable"}\}_{i \in \mathbb {N}})^\circ $ .

Proof Let $s \colon \mathbb {N} \to \mathbb {N}$ be strictly increasing, $i \in \mathbb {N}$ , and let $\phi (u^i, v^i, W^{i+2})$ be the following ${\mathcal {L}_{\mathrm {TST}}}$ -formula:

Let $\chi (W^{i+2})$ be the following ${\mathcal {L}_{\mathrm {TST}}}$ -formula:

$$ \begin{align*} &\forall u^i, v^i (\phi(u^i, v^i, W^{i+2}) \vee \phi(v^i, u^i, W^{i+2}) \vee u^i = v^i) \\ & \quad\wedge \forall u^i, v^i, w^i (\phi(u^i, v^i, W^{i+2}) \wedge \phi(v^i, w^i, W^{i+1}) \to \phi(u^i, w^i, W^{i+2})) \\ &\quad\wedge \forall x^{i+1} \exists u^i (u^i \varepsilon x^{i+1} \wedge \forall v^i (v^i \varepsilon x^{i+1} \wedge u^i \neq v^i \to \phi(u^i, v^i, W^{i+2}))). \end{align*} $$

For all $W \in A_{s(i+2)}$ , we have $\mathscr {A} \models \chi ^s(W)$ iff

$$ \begin{align*} &\forall u, v \in A_{s(i)} (\phi^{\mathscr{A},s}(u, v, W) \vee \phi^{\mathscr{A},s}(v, u, W) \vee (u)_{s(i)} = (v)_{s(i)}) \\ &\quad\wedge \forall u, v, w \in A_{s(i)} (\phi^{\mathscr{A},s}(u, v, W) \wedge \phi^{\mathscr{A},s}(v, w, W) \to \phi^{\mathscr{A},s}(u, w, W)) \\ &\quad\wedge \forall x \in A_{s(i+1)} \exists u \in A_{s(i)} ((u)_{s(i)} \in (x)_{s(i)+1} \\ &\quad\wedge \forall v \in A_{s(i)} ((v)_{s(i)} \in (x)_{s(i)+1} \wedge (u)_{s(i)} \neq (v)_{s(i)} \to \phi^{\mathscr{A},s}(u, v, W))), \end{align*} $$

where $\phi ^{\mathscr {A},s}(u, v, W)$ is the following ${\mathcal {L}_{\mathrm {ZF}}}$ -formula:

$$\begin{align*}\exists x \in A_{s(i+1)} ((x)_{s(i+1)} \in (W)_{s(i+1)+1} \wedge (u)_{s(i)} \in (x)_{s(i)+1} \wedge (v)_{s(i)} \notin (x)_{s(i)+1}). \end{align*}$$

Let $\psi (u,v)$ be an ${\mathcal {L}_{\mathrm {ZF}}}$ -formula that defines some well-ordering of L. Let

$$ \begin{align*} &W' = \{ (x)_{s(i+1)} : x \in A_{s(i+1)} \\ &\quad\wedge \exists u \in A_{s(i)} \forall v \in A_{s(i)} ((v)_{s(i)} \in (x)_{s(i)+1} \leftrightarrow \psi((u)_{s(i)}, (v)_{s(i)}))\}. \end{align*} $$

We know that $W' \in L_{\alpha _{s(i+1)+1}} \cap {\mathscr {P}}(X_{s(i+1)}) = X_{s(i+1)+1}$ , so there exists some $W \in A_{s(i+2)}$ such that $(W)_{s(i+1)+1} = W'$ . Notice that $W'$ is the set of all $(x)_{s(i+1)}$ for which $x \in A_{s(i+1)}$ and $(x)_{s(i)+1}$ is an initial segment of the well-ordering defined by $\psi $ restricted to $X_{s(i)}$ . Therefore, W witnesses that $\mathscr {A} \models \chi ^s(W)$ . Now, let

$$\begin{align*}R' = \{ (z)_{s(i+2)} : z \in A_{s(i+2)} \wedge \exists u,v \in A_{s(i)} (\mathrm{Pair}^{\mathscr{A},s}_i(u,v,z) \wedge \phi^{\mathscr{A},s}(u,v,W)) \}. \end{align*}$$

We have that $R' \in L_{\alpha _{s(i+2)+1}} \cap {\mathscr {P}}(X_{s(i+2)}) = X_{s(i+2)+1}$ , so there exists an $R \in A_{s(i+3)}$ such that $(R)_{s(i+2)+1} = R'$ . It is easy to see that $\mathscr {A} \models \chi ^s(W)$ implies $\mathscr {A} \models (\text {"}R$ is a well-ordering of $V^{i+1}\text {"})^s$ . Hence, $\mathscr {A} \models (\text {"}V^{i+1} \text { is well-orderable"})^s$ .

Next, we show that in $\mathscr {A}$ every universe is cantorian (i.e., it is equinumerous to the set of singletons). This essentially follows from the fact that for all $i, j \in \mathbb {N}$ , there are functions in $\mathscr {A}$ witnessing that $|X_i| = \kappa = |\iota "X_j|$ .

Lemma 2.3. $\mathscr {A} \models (\{\text {"}V^{i+2} \text { is cantorian"}\}_{i \in \mathbb {N}})^\circ $ .

Proof Let $s \colon \mathbb {N} \to \mathbb {N}$ be strictly increasing, and let $i \in \mathbb {N}$ . We show that $\mathscr {A} \models (\text {"}V^{i+2} \text { is cantorian"})^s$ , i.e., that

$$ \begin{align*} \mathscr{A} \models (\exists g^{i+4}(\text{"}g\text{ is a bijection from }V^{i+2}\text{ to }\iota"V^{i+1}\text{"}))^s. \end{align*} $$

An ${\mathcal {L}_{\mathrm {TST}}}$ -sentence that expresses the statement

$$\begin{align*}\exists g^{i+4}(\text{"}g\text{ is a bijection from }V^{i+2}\text{ to }\iota"V^{i+1}\text{"}) \end{align*}$$

is the following:

$$ \begin{align*} &\exists g^{i+4} ( \forall z^{i+3} (z^{i+3} \varepsilon g^{i+4} \to \exists u^{i+1}, v^{i+1} (\mathrm{Pair}_{i+1}(u^{i+1}, v^{i+1}, z^{i+3}))) \\ &\wedge \forall u^{i+1}, v^{i+1}, w^{i+1}, x^{i+3}, y^{i+3} (\mathrm{Pair}_{i+1}(u^{i+1}, v^{i+1}, x^{i+3}) \wedge \mathrm{Pair}_{i+1}(u^{i+1}, w^{i+1}, y^{i+3}) \\ &\wedge x^{i+3} \varepsilon g^{i+4} \wedge y^{i+3} \varepsilon g^{i+4} \to v^{i+1} = w^{i+1}) \\ &\wedge \forall u^{i+1}, v^{i+1}, w^{i+1}, x^{i+3}, y^{i+3} (\mathrm{Pair}_{i+1}(u^{i+1}, w^{i+1}, x^{i+3}) \wedge \mathrm{Pair}_{i+1}(v^{i+1}, w^{i+1}, y^{i+3}) \\ &\wedge x^{i+3} \varepsilon g^{i+4} \wedge y^{i+3} \varepsilon g^{i+4} \to u^{i+1} = v^{i+1}) \\ &\wedge \forall u^{i+1} \exists v^{i+1}, z^{i+3} (\mathrm{Pair}_{i+1}(u^{i+1}, v^{i+1}, z^{i+3}) \wedge z^{i+3} \varepsilon g^{i+4} \\ &\wedge \exists a^i \forall b^i (b^i \varepsilon v^{i+1} \leftrightarrow b^i = a^i)) \\ &\wedge \forall v^{i+1} (\exists a^i \forall b^i (b^i \varepsilon v^{i+1} \leftrightarrow b^i = a^i) \\ &\to \exists u^{i+1}, z^{i+3} (\mathrm{Pair}_{i+1}(u^{i+1}, v^{i+1}, z^{i+3}) \wedge z^{i+3} \varepsilon g^{i+4}))). \end{align*} $$

We therefore have that $\mathscr {A} \models (\text {"}V^{i+2} \text { is cantorian"})^s$ iff

$$ \begin{align*} &\exists g \in A_{s(i+4)} ( \forall z \in A_{s(i+3)} ((z)_{s(i+3)} \in (g)_{s(i+3)+1} \\ &\to \exists u, v \in A_{s(i+1)} (\mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, v, z))) \\ &\wedge \forall u, v, w \in A_{s(i+1)} \forall x, y \in A_{s(i+3)} (\mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, v, x) \wedge \mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, w, y) \\ &\wedge (x)_{s(i+3)} \in (g)_{s(i+3)+1} \wedge (y)_{s(i+3)} \in (g)_{s(i+3)+1} \to (v)_{s(i+1)} = (w)_{s(i+1)}) \\ &\wedge \forall u, v, w \in A_{s(i+1)} \forall x, y \in A_{s(i+3)} (\mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, w, x) \wedge \mathrm{Pair}^{\mathscr{A},s}_{i+1}(v, w, y) \\ &\wedge (x)_{s(i+3)} \in (g)_{s(i+3)+1} \wedge (y)_{s(i+3)} \in (g)_{s(i+3)+1} \to (u)_{s(i+1)} = (v)_{s(i+1)}) \\ &\wedge \forall u \in A_{s(i+1)} \exists v \in A_{s(i+1)} \exists z \in A_{s(i+3)} (\mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, v, z) \wedge (z)_{s(i+3)} \in (g)_{s(i+3)+1} \\ &\wedge \exists a \in A_{s(i)} \forall b \in A_{s(i)} ((b)_{s(i)} \in (v)_{s(i)+1} \leftrightarrow (b)_{s(i)} = (a)_{s(i)})) \\ &\wedge \forall v \in A_{s(i+1)} (\exists a \in A_{s(i)} \forall b \in A_{s(i)} ((b)_{s(i)} \in (v)_{s(i)+1} \leftrightarrow (b)_{s(i)} = (a)_{s(i)}) \\ &\to \exists u \in A_{s(i+1)} \exists z \in A_{s(i+3)} (\mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, v, z) \wedge (z)_{s(i+3)} \in (g)_{s(i+3)+1}))). \end{align*} $$

Let

$$ \begin{align*} &g' = \{ (z)_{s(i+3)} : z \in A_{s(i+3)} \wedge \exists u, v \in A_{s(i+1)} ( \mathrm{Pair}^{\mathscr{A},s}_{i+1}(u, v, z) \\ &\wedge \exists \alpha < \kappa ( (u)_{s(i+1)} = f_{s(i+1)}(\alpha) \wedge (v)_{s(i)+1} = \{f_{s(i)}(\alpha)\})) \}. \end{align*} $$

We have that $g' \in L_{s(i+3)+1} \cap {\mathscr {P}}(X_{s(i+3)}) = X_{s(i+3)+1}$ , so there is a $g \in A_{s(i+4)}$ such that $(g)_{s(i+3)+1} = g'$ . Clearly, g witnesses that

$$\begin{align*}\mathscr{A} \models (\exists g^{i+4} (\text{"}g \text{ is a bijection from }V^{i+2}\text{ to }\iota"V^{i+1}\text{"}))^s.\\[-36pt] \end{align*}$$

We have shown that

$$\begin{align*}\mathrm{TTTP} + (\{\text{"}V^{i+1} \text{ is well-orderable"}\}_{i \in \mathbb{N}})^\circ + (\{\text{"}V^{i+2} \text{ is cantorian"}\}_{i \in \mathbb{N}})^\circ \end{align*}$$

is consistent. By slightly modifying Holmes’ proof for the equiconsistency of $\mathrm {TTT}$ and $\mathrm {NF}$ (see [3]), we can now prove the following proposition.

Proposition 2.4. $\mathrm {NFP} + \text {"}V\text { is well-orderable"} + \text {"}V\text { is cantorian"}$ is consistent.

Proof Let $\Delta = \{\sigma _1, \ldots , \sigma _n\}$ be a finite subset of

$$\begin{align*}\mathrm{TTTP} + (\{\text{"}V^{i+1} \text{ is well-orderable"}\}_{i \in \mathbb{N}})^\circ + (\{\text{"}V^{i+2} \text{ is cantorian"}\}_{i \in \mathbb{N}})^\circ. \end{align*}$$

Let $m \in \mathbb {N}$ be such that all variables appearing in $\sigma _1, \ldots , \sigma _n$ have types that are less or equal to m. For each $X \subseteq \mathbb {N}$ such that $|X| = m+1$ , let $s_X$ be the unique strictly increasing function from $\{0, \ldots , m\}$ to X. Let $F \colon [\mathbb {N}]^{m+1} \to 2^n$ such that for all $X \in [\mathbb {N}]^{m+1}$ , $F(X) = (\delta _1, \ldots , \delta _n)$ , where for each $1 \leq i \leq n$ , $\delta _i {\kern-1.2pt}={\kern-1.2pt} 1$ iff $\mathscr {A} {\kern-1.2pt}\models{\kern-1.2pt} (\sigma _i)^{s_X}$ . By Ramsey’s theorem, there exists some infinite $H {\kern-1.2pt}\subseteq{\kern-1.2pt} \mathbb {N}$ such that H is homogeneous for F. Let $H {\kern-1.5pt}={\kern-1.2pt} \{h_0, h_1, \ldots \}$ , where $h_0 {\kern-1.2pt}<{\kern-1.2pt} h_1 {\kern-1.2pt}<{\kern-1.2pt} \cdots $ , and let $\mathscr {B} = (A_{h_0}, A_{h_1}, \ldots , \{\varepsilon ^{\mathscr {A}}_{h_i,h_{i+1}}\}_{i \in \mathbb {N})})$ . It is easy to see that $\mathscr {B}$ is a model of $\mathrm {TSTP} {\kern-1.2pt}+{\kern-1.2pt} \{\text {"}V^{i+1}\text { is well-orderable"}\}_{i \in \mathbb {N}} {\kern-1.2pt}+{\kern-1.2pt} \{\text {"}V\text { is cantorian"}\}_{i \in \mathbb {N}} {\kern-1.2pt}+{\kern-1.2pt} \{ \sigma {\kern-1.2pt}\leftrightarrow{\kern-1.2pt} \sigma ^+ {\kern-1.5pt}: \sigma {\kern-1.2pt}\in{\kern-1.2pt} \Delta \}$ , where $\sigma ^+$ is the sentence we get if we raise the type of every variable of $\sigma $ by one. By Compactness, it follows that $\mathrm {TSTP} + \{\text {"}V^{i+1}\text { is well-orderable"}\}_{i \in \mathbb {N}} + \{\text {"}V^{i+2}\text { is cantorian"}\}_{i \in \mathbb {N}} + \{ \sigma \leftrightarrow \sigma ^+ : \sigma \text { is a sentence } \}$ is consistent. Therefore, by Specker’s results on ambiguity (see [2] or [7]), it follows that there is a model of $\mathrm {TSTP} + \{\text {"}V^{i+1}\text { is well-orderable"}\}_{i \in \mathbb {N}} + \{\text {"}V^{i+2}\text { is cantorian"}\}_{i \in \mathbb {N}}$ with a type shifting automorphism, which means that there is a model of $\mathrm {NFP} + \text {"}V\text { is well-orderable"} + \text {"}V\text { is cantorian."}$

Although $\mathrm {NFP}$ is consistent with a very strong choice principle like the one above, it is inconsistent with some other forms of choice. For example, let $\mathrm {AC}$ be the statement “for any set x of non-empty pairwise disjoint sets, there exists a choice set z, i.e., a set that has exactly one element in common with each element of x,” which can be can be expressed formally as the following ${\mathcal {L}_{\mathrm {NF}}}$ -sentence:

Theorem 2.5. $\mathrm {NFP} \vdash \neg \mathrm {AC}$ .

Proof As Crabbé observed in [1], $\mathrm {NFP} + \forall x (x \subseteq \iota "V \to \exists y (x = \iota "y)) = \mathrm {NF}$ . But, in $\mathrm {NFP}$ , $\mathrm {AC}$ implies $\forall x (x \subseteq \iota "V \to \exists y (x = \iota "y))$ because if x is a set of singletons and y is a choice set for x, then $x = \iota "y$ . So, $\mathrm {NFP} + \mathrm {AC} = \mathrm {NF}$ , which means that $\mathrm {NFP} \vdash \neg \mathrm {AC}$ .

Note. Notice of course that the above form of choice is impredicative, and therefore not really suitable for a predicative theory. A more sensible and unproblematic statement in this setting would be the following: “for any set x of non-empty pairwise disjoint sets, there exists a set of singletons z, where every element of a singleton in z belongs to exactly one element of x” (notice that in this version, z has the same relative type as x).

3 Conclusion

We described a simple way of constructing a model of $\mathrm {NFP}$ using L. We also showed that there are properties of L that can be transferred naturally to this model. We have chosen to present just two such properties that are inconsistent with $\mathrm {NF}$ , but there are others. It would be nice to have a more general result on what kind of properties can be transferred though.

Question 1. What properties of L can be transferred to our model of $\mathrm {NFP}$ ?

The construction of our model seems to be quite flexible. For example, we could modify the definition of $\alpha _n$ so that more sets are included in $X_{\alpha _n}$ . This could lead us to stronger consistent extensions of $\mathrm {NFP}$ . For example, the following question seems promising.

Question 2. Can this construction be modified so that we get a model of $\mathrm {NFP}$ satisfying some weak version of Union?

Finding consistency proofs for subtheories of Tangled Type Theory is really important for understanding the problem of the consistency of $\mathrm {NF}$ . On the other hand, the opposite direction is equally interesting: if we assume the consistency of $\mathrm {NF}$ (see [5]), and therefore of Tangled Type Theory, we may get new results in other areas of set theory. For example, as we have shown, there seems to be a connection between models of Tangled Type Theory and L.

Question 3. Does the consistency of Tangled Type Theory have any implications for L?