Why Predicative Sets?

Abstract

The predicativist program for the foundations of mathematics, initiated by Poincaré and first developed by Weyl, seeks to establish certainty in mathematics without revolutionizing it. The program was later extensively pursued by Feferman, who developed proofs systems for predicative mathematics, and showed that a very large part of classical analysis can be developed within them. Both Weyl and Feferman worked within type-theoretic frameworks. In contrast, set theory is almost universally accepted now as the foundational theory in which the whole of mathematics can and should be developed. We explain how to reconstruct Weyl’s ideas and system within the set-theoretical framework, and indicate the advantages that this approach to predicativity and to set theory has from both the foundational as well as the computational points of views.

Appendices

Appendix 1. WA: Weyl’s System in [27]

We use $\sigma$ and $\tau$ as metavariables for types, t, s as metavariables for terms, and $\varphi ,\psi$ as metavariables for formulas. We also employ x, y, z, w as general variables for objects, n, k, m as variables for objects of type N, f, g as variables for objects of types of functions, X, Y, Z for objects of types of sets. We let $\overline{\sigma }=\sigma _1\times \cdots \times \sigma _k$, $\overline{\tau }=\tau _1\times \cdots \times \tau _n$, $\mathbf {x}=x_1,\ldots ,x_n$, $\mathbf {w}=w_1,\ldots ,w_n$, $\mathbf {y}=y_1,\ldots ,y_k$, $\mathbf {z}=z_1,\ldots ,z_k$, $\mathbf {f}=f_1,\ldots ,f_m$, $\mathbf {X}=X_1,\ldots ,X_m$, $\forall x\ldots =\lnot \exists x\lnot \ldots$, $\exists x:\sigma \ldots =\exists x^{\sigma }\ldots$, $\forall \mathbf {z}:\overline{\sigma }\ldots =\forall z_1:\sigma _1\cdots \forall z_k:\sigma _k\ldots$.

A.1 Language

• Types

  • 1. N is a basic type.

  • 2. If $\sigma _1,\ldots ,\sigma _k$ and $\tau _1,\ldots ,\tau _n$ are types, where $k\ge 0$ and $n\ge 1$, then $(\sigma _1\times \cdots \times \sigma _k)\rightarrow S(\tau _1\times \cdots \times \tau _n)$ is a type.

• Terms and their type(s)

  • 1. $x^{\sigma }:\sigma$ whenever $x^{\sigma }$ is a variable of type $\sigma$.¹¹ (We assume an infinite supply of variables $x^{\sigma }$ for each type $\sigma$.)

  • 2. $f(t_1,\ldots ,t_k):S(\overline{\tau })$ in case $f:\overline{\sigma }\rightarrow S(\overline{\tau })$ and $t_i:\sigma _i$ for $1\le i\le k$.

  • 3. $\{(x_1,\ldots , x_n)\mid \psi \}:S(\overline{\tau })$ whenever $n\ge 1$, $x_i:\tau _i$ for $1\le i\le n$, and $\psi$ is a delimited formula.

  • 4. $\lambda y_1,\ldots ,y_k.t:\overline{\sigma }\rightarrow S(\overline{\tau })$ in case $t:S(\overline{\tau })$, and $y_i:\sigma _i$ for $1\le i\le k$.

  • 5. $IT^i_m(f_1,\ldots ,f_m):\mathbf{N}\times \overline{\sigma }\times S(\overline{\tau })^m \rightarrow S(\overline{\tau })$ in case $m>0$, and for $1\le i\le m$, either $f_i:\overline{\sigma }\times S(\overline{\tau })\rightarrow S(\overline{\tau })$ or $f_i:\mathbf{N}\times \overline{\sigma }\times S(\overline{\tau })\rightarrow S(\overline{\tau })$.

• Delimited Formulas (d.f.)

  • 1. If $t:\mathbf{N}$ and $s:\mathbf{N}$, then Succ(t, s) is a delimited formula.

  • 2. If $t:\mathbf{N}$ and $s:\mathbf{N}$, then $t=s$ is a delimited formula.

  • 3. If $t_1,\ldots ,t_n$ are terms of types $\tau _1,\ldots ,\tau _n$ respectively, and $s:S(\overline{\tau })$, then $(t_1,\ldots ,t_n)\in s$ is a delimited formula.

  • 4. If $\varphi$ and $\psi$ are s. f. then so are $\lnot \varphi$, $(\varphi \wedge \psi )$ and $(\varphi \vee \psi )$.

  • 5. If x is a variable of type $\mathbf{N}$, and $\varphi$ is a d. f. then so is $\exists x\varphi$.

• Formulas

  • 1. If $t:\mathbf{N}$ and $s:\mathbf{N}$, then Succ(t, s) is a formula.

  • 2. If t and s are terms of the same type then $t=s$ is a formula.

  • 3. If $t_1,\ldots ,t_n$ are terms of types $\tau _1,\ldots ,\tau _n$ respectively, and $s:S(\overline{\tau })$, then $(t_1,\ldots ,t_n)\in s$ is a formula.

  • 4. If $\varphi$ and $\psi$ are formulas then so are $\lnot \varphi$, $(\varphi \wedge \psi )$ and $(\varphi \vee \psi )$.

  • 5. If x is a variable and $\varphi$ is a formula, then $\exists x\varphi$ is a formula.

A.2 Proof System

• Logic: Many-sorted first order logic with variable-binding terms operators, and with equality in all sorts (i.e. types).

• Axioms:

  • Comprehension Axioms

    • $\forall \mathbf {w}.(\mathbf {w})\in \{(\mathbf {x})\mid \psi \}\leftrightarrow \psi [\mathbf {w}/\mathbf {x}]$

    • $\forall \mathbf {z}.(\lambda \mathbf {y}.t)(\mathbf {z})=t[\mathbf {z}/\mathbf {y}]\}$

  • Extensionality Schema

    • $\forall X:S(\overline{\tau })\forall Y:S(\overline{\tau }). X=Y \leftrightarrow \forall \mathbf {w}:\overline{\tau }.\mathbf {w}\in X\leftrightarrow \mathbf {w}\in Y$

    • $\forall f:\overline{\sigma }\rightarrow S(\overline{\tau }) \forall g:\overline{\sigma }\rightarrow S(\overline{\tau }). f=g \leftrightarrow \forall \mathbf {z}:\overline{\sigma }. f(\mathbf {z})=g(\mathbf {z})$

  • The standard axioms for Succ

    • $\exists ! n\forall k.\lnot Succ(k,n)$

    • $\forall k \exists ! n. Succ(k,n)$

    • $\forall k \forall m \forall n. Succ(k,n)\wedge Succ(m,n)\rightarrow k=m$

  • Induction Schema

    $\psi \{0/n\}\wedge (\forall n\forall k.Succ(n,k)\wedge \psi \rightarrow \psi \{k/n\})\rightarrow \forall n\psi$

  • Axioms Schemas for iteration

    For each $1\le i\le m$:

    • $\forall {\mathbf {z}}\forall {\mathbf {f}} \forall {\mathbf {X}}. IT^i_m({\mathbf {f}})(1,{\mathbf {z}},{\mathbf {X}})= f_i([1,]{\mathbf {z}},{\mathbf {X}})$

    • $\forall n\forall k\forall {\mathbf {z}}\forall {\mathbf {f}}\forall {\mathbf {X}}. Succ(n,k)\rightarrow IT^i_m({\mathbf {f}})(k,{\mathbf {z}},{\mathbf {X}})=$$IT^i_m({\mathbf {f}})(n,{\mathbf {z}}, f_1([k,]{\mathbf {z}},{\mathbf {X}}), \ldots , f_m([k,]{\mathbf {z}},{\mathbf {X}}))$

      Where depending on the type of $f_i$ , $f_i([k,]{\mathbf {z}},{\mathbf {X}})$ means either $f_i({\mathbf {z}},{\mathbf {X}})$ or $f_i(k,{\mathbf {z}},{\mathbf {X}})$, and similarly with $f_i([1,]{\mathbf {z}},{\mathbf {X}})$.

Appendix 2: The Formal System $\mathsf {PZF}$

Language

The language $\mathcal{L}_{PZF}$ of $\mathsf {PZF}$ is defined by a simultaneous recursion.

• Predicates and Operations

  • $=$ and $\in$ are binary predicates.

  • If $\varphi$ is a formula such that $\varphi \succ _{PZF}\emptyset$, and $Fv(\varphi )=\{x_1,\ldots ,x_n\}$ where $n>0$, then $[(x_1,\ldots ,x_n)\mid \varphi ]$ is an n-ary predicate.

  • If t is a term such that $Fv(t)=\{y_1,\ldots ,y_k\}$, then $\lambda y_1,\ldots ,y_k.t$ is a k-ary operation.

• Terms:

  • Every variable is a term.

  • If $\varphi \succ _{PZF}\{x\}$, then $\{x\mid \varphi \}$ is a term.

  • If F is a k-ary operation, and $t_1,\ldots ,t_k$ are terms, then $F(t_1,\ldots ,t_k)$ is a term.

• Formulas:

  • If P is an n-ary predicate, then $P(t_1,\ldots ,t_n)$ is an atomic formula whenever $t_1,\ldots ,t_n$ are terms.

  • If $\varphi$ and $\psi$ are formulas, and x is a variable, then $\lnot \varphi$, $(\varphi \wedge \psi )$, $(\varphi \vee \psi )$, and $\exists x\varphi$ are formulas. ($\forall x\varphi$ and $\varphi \rightarrow \psi$ are taken as abbreviations for $\lnot \exists x\lnot \varphi$ and $\lnot (\varphi \wedge \lnot \psi )$, respectively.)

  • If $\varphi$ is a formula, t and s are terms, and x and y are distinct variables, then $(TC_{x,y}\varphi )(t,s)$ is a formula, and

    $$Fv((TC_{x,y}\varphi )(t,s))=(Fv(\varphi )-\{x,y\})\cup Fv(t) \cup Fv(s)$$

• The Safety Relation $\succ _{PZF}$:

  • ($\in$) $x\in t\succ _{PZF}\{x\}$ if $x\not \in Fv(t)$.

  • (At) $\varphi \succ _{PZF}\emptyset$ if $\varphi$ is atomic.

  • (=)] $\varphi \succ _{PZF}\{x\}$ if $\varphi \in \{x\ne x, x=t,t=x\}$, and $x\not \in Fv(t)$.

  • ($\lnot$) $\lnot \varphi \succ _{PZF}\emptyset$ if $\varphi \succ _{PZF}\emptyset$.

  • ($\vee$) $\varphi \vee \psi \succ _{PZF}X$ if $\varphi \succ _{PZF}X$ and $\psi \succ _{PZF}X$.

  • ($\wedge$) $\varphi \wedge \psi \succ _{PZF}X\cup Y$ if $\varphi \succ _{PZF}X$, $\psi \succ _{PZF}Y$, and $Y\cap Fv(\varphi )=\emptyset$.

  • ($\exists$) $\exists y \varphi \succ _{PZF}X-\{y\}$ if $y\in X$ and $\varphi \succ _{PZF}X$.

  • (TC) $(TC_{x,y}\varphi )(x,y)\succ _{PZF}X$ if $\varphi \succ _{PZF}X\cup \{x\}$, or $\varphi \succ _{PZF}X\cup \{y\}$.

Logic and Axioms

• Logic: Classical $\mathcal {AL}$ with variable-binding terms operators, and equality.

• Axioms:

  • Extensionality: $\forall z (z\in x \leftrightarrow z\in y)\rightarrow x=y$

  • Comprehension: The universal closures of formulas of the forms:

    • $x\in \{x\mid \varphi \}\leftrightarrow \varphi$

    • $[(x_1,\ldots ,x_n)\mid \varphi ](t_1,\ldots ,t_n) \leftrightarrow \varphi \{t_1/x_1,\ldots ,t_n/x_n\}$

    • $(\lambda y_1,\ldots ,y_k.t)(s_1,\ldots ,s_k)=t\{s_1/y_1,\ldots ,s_k/y_k\}$

  • $\in$ -induction: $(\forall x(\forall y(y\in x\rightarrow \varphi \{y/x\})\rightarrow \varphi ))\rightarrow \forall x\varphi$