Arnon Avron

Weyl’s ‘Das Kontinuum’ — 100 years later

Orevkov’80 Conference St. Petersburg Days of Logic and Computability V April 2020 Prologue

All platonists are alike; each anti-platonist is unhappy in her/his own way...

My aim in this talk is first of all to present Weyl’s views and system, at the time he wrote “Das Kontinuum” (exactly 100 years ago). Then I’ll try to describe mine, which I believe are rather close to Weyl’s original ideas (but still different). Weyl’s Goals

“I shall show that the house of analysis is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility.”

“I would like to be understood . . . by all students who have become acquainted with the currently canonical and al- legedly ‘rigorous’ foundations of analysis.”

“In spite of Dedekind, Cantor, and Weierstrass, the great task which has been facing us since the Pythagorean dis- covery of the irrationals remains today as unfinished as ever” Weyl and P´olya’s Wager in 1918

Within 20 years P´olya and the majority of representative mathematicians will admit that the statements

1 Every bounded set of reals has a precise supremum 2 Every infinite set of numbers contains a denumerable subset contain totally vague concepts, such as “number,” “set,” and “denumerable,” and therefore that their truth or fal- sity has the same status as that of the main propositions of Hegel’s natural philosophy. However, under a natural interpretation, (1) and (2) will be seen to be false. What Has Weyl Rejected, and Why?

He rejected the “completely vague concept of function which has become canonical in analysis since Dirichlet and, together with it, the prevailing concept of set.”

He believed that this “vague concept of set and function . . . finds itself caught in a vicious circle.” The Ideal Calculus for Sets

Extensionality:

∀z(z ∈ x ↔ z ∈ y) → x = y

The Comprehension Schema:

∀y(y ∈{ x | ϕ}↔ ϕ[y/x])

Ideal, but inconsistent! Predicativity, Impredicativity, and Poincar´eVCP

The inconsistency of the ideal calculus implies that not every formula ϕ can be used for constructing legitimate set terms of the form {x | ϕ}. Poincar´eand Russel called a formula that (in their opinion) can be so used predicative, and a formula that cannot impredicative.

They (and then Weyl) Usually characterized an impredicative definition as a definition that violates the Vicious Circle Principle (VCP):

‘Whatever involves all of a collection must not be one of the collection.’ [Principia Mathematica 1910] Weyl and the VCP

“It would be meaningless to include among these principles an assertion such as the following: If A is a property of properties, then one may form that property PA which belongs to an object x if and only if there is a property constructed by means of these principles which belongs to x and itself possesses the property A. That would be a blatant vicious circle; yet our current version of analysis commits this error and I consider this ground for censure”.

S PA = {x ∈ S | ∃X .X ∈ 2 ∧ A(X ) ∧ x ∈ X } An Example: The LUB principle

In analysis, the least upper bound (sup) of a bounded set A of reals is defined as the minimal element of the set of upper bounds of A. This is an impredicative definition. When real numbers are taken as Dedekind cuts, then the existence of the sup(A) is shown by defining (or characterizing it) as:

sup(A) = {q ∈ Q | ∃X .X ∈ 2Q ∧ X ∈ A ∧ q ∈ X }

Again, this involves an impredicative definition. When Should an Impredicative Definition Be Rejected?

There are certainly impredicative definitions that are perfectly acceptable: The tallest man in the room The least prime number These definitions just select an object from a given fully-determined (or ‘closed’) collection.

An impredicative definition is illegitimate if it tries to “select” an object from an indeterminate (or ‘open’) collection. In particular: if the definition is viewed as creating the object it defines. Weyl’s Principles: Sets

Sets (and functions) are created only by definitions.

“No one can describe an infinite set other than by indi- cating properties which are characteristic of the elements of the set. . . . The notion that an infinite set as a “gath- ering” brought together by infinitely many individual arbi- trary acts of selection, assembled and then surveyed as a whole by consciousness, is nonsensical;” Weyl’s Principles: Predicativity Via Types

Weyl allows only predicative definitions of sets. Like Russel, this predicativity is ensured using a combination of two independent means, one of which is the use of types (or “categories”).

Each object and relation between objects has a specific type.

Some of the types are basic. Each basic type is “a complete system of definite self-existent objects”. In addition, in basic types also the basic accepted relations should be “definite”.

The other types are types of sets and functions which are derived from the basic ones using the rules of definition. These types are incomplete systems of objects. Weyl’s Principles: Predicativity Via Safe Formulas

The other measure Weyl took in order to ensure predicativity is to allow only the use of safe (or “delimited”) formulas in the definitions of sets and functions.

For Weyl safe formulas are those in which the use of quantification and equality is confined only to basic types. Weyl’s Principles: The Natural Numbers

The natural numbers sequence is a basic well-understood, mathematical concept, and should be taken as a basic type.

“[T]he idea of iteration, i.e., of the sequence of the natural numbers, is an ultimate foundation of mathematical thought” “Our grasp of the basic concepts of set theory depends on a prior intuition of iteration and of the sequence of natural numbers.” [P. 24]

“A single basic relation, whose meaning is immediately exhibited, underlies this category — namely, the relation Succ(x, y) which holds between two natural numbers x, y when y is the immediate successor of x.” [P. 25] Weyl’s Principles: Real numbers

[T]he real numbers are certain four-dimensional sets of natural numbers. The collection of real numbers is open: “If we regard the principles of definition as an “open” sys- tem, i.e., if we reserve the right to extend them when necessary by making additions, then in general the ques- tion of whether a given function is continuous must also remain open. For a function which, within our current sys- tem, is continuous can lose this property if our principles of definition are expanded and, accordingly, the real numbers “presently” available are joined by others.” [P. 87] Weyl’s Notion of Function

The only sort of functions as objects which are allowed by Weyl are those which have explicit definitions of the form:

λy1,..., yk .{hx1,..., xni | ϕ}

The domain of such a function is some product of types, and its range is a type of sets. Thus operations on N (like addition) are officially defined as relations, not as functions. Functions are always defined on their full domain. Thus a function from R to R is a function from Set(N4) to Set(N4) with the property that for every x ∈ R its value is in R. (The collection of such functions is neither a set nor a type.)

Weyl writes: “once we become aware of this concept of function, we also immediately grasp its significance”. Some Examples of Weyl’s Idea

Procedures for solving equations and inequalities (with parameters). E.g. Asin(nx) + Bcos(nx) = C.

Constructions of locies in Euclidean geometry.

Queries in logic programming.

Queries in relational databases. Weyl’s Formal System: Types

1 N is a (basic) type.

2 If τ1, . . . , τn are types (n > 0), then so is Set(τ1 × · · · × τn).

3 If σ1, . . . , σk and τ1, . . . , τn are types, where k > 0 and n > 0, then( σ1 × · · · × σk ) → Set(τ1 × · · · × τn) is a type. Weyl’s Formal System: Terms

Notation: σ = σ1 × · · · × σk , τ = τ1 × · · · × τn. 1 xσ : σ

2 Let n > 0, and let ψ be a safe formula such that Fv[ψ] ⊇ {x1,..., xn}, where xi : τi (1 ≤ i ≤ n). Then

{hx1,..., xni | ψ} : Set(τ)

3 Let k > 0, and let t : Set(τ) be a term such that Fv[t] ⊇ {y1,..., yk }, where yj : σj (1 ≤ j ≤ k). Then

λy1,..., yk .t : σ →Set(τ)

4 If s : σ → Set(τ) and ti : σi (i = 1,..., k), then s(t1,..., tk ): Set(τ).

5 If t : Set(τ) → Set(τ) then IT [t]: N × Set(τ) → Set(τ). A Natural Extension of Weyl’s System

Types:3 If σ1, . . . , σk and τ are types, where k ≥ 1, then (σ1 × · · · × σk ) → τ is a type.

Terms:3 Let k ≥ 1, and let t : τ be a term such that Fv[t] ⊇ {y1,..., yk }, where yj : σj (1 ≤ j ≤ k). Then λy1,..., yk .t : σ → τ. 4 If s : σ → τ and ti : σi (i = 1,..., k), then s(t1,..., tk ): τ. 5 If t : τ → τ then IT [t]: N × τ → τ. Weyl’s Formal System: Safe Formulas

The set of terms and the set of safe formulas are simultaneously defined using a mutual recursion.

1 If t : N and s : N then SUCC(t, s) is a safe formulas. 2 If t and s are terms of type N then t = s is a safe formula.

3 If ti : τi for i = 1,..., n and s : Set(τ1 × · · · × τn), then ht1,..., tni ∈ s is a safe formula.

4 If ϕ and ψ are safe formulas then so are ¬ϕ,( ϕ ∧ ψ),( ϕ ∨ ψ).

5 If x : N is a variable and ϕ is a safe formula, then so is ∃xϕ. Weyl’s Formal System: Formulas

1 If t : N and s : 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 ti : τi for i = 1,..., n and s : Set(τ1 × · · · × τn), then ht1,..., tni ∈ s is a formula.

4 If ϕ and ψ are formulas then so are ¬ϕ,( ϕ ∧ ψ) and( ϕ ∨ ψ).

5 If x is an arbitrary variable and ϕ is a formula, then so is ∃xϕ. Weyl’s Formal System: Axioms for Natural Numbers

Peano’s Axioms: ∃!n∀k.¬Succ(k, n) ∧ ∀k∃!n.Succ(k, n) ∀k∀m∀n.Succ(k, n) ∧ Succ(m, n) → k = m Induction Schema: ψ{1/n} ∧ (∀n∀k.Succ(n, k) ∧ ψ → ψ[k/n]) → ∀nψ

Axiom Schemas for iteration: ∀f .IT [f ](1, X ) = f (X ) ∀n∀k∀f .Succ(n, k) → IT [f ](k, X ) = IT [f ](n, f (X )) Weyl’s Formal System: Axioms for Sets and Functions

Comprehension Axiom Schemas ∀w~ .w~ ∈ {~x | ψ} ↔ ψ[w~ /~x]

∀~z.(λ~y.t)(~z) = t[~z/~y]}

Extensionality Axiom Schemas ∀X ∀Y (X = Y ↔ ∀w~ .w~ ∈ X ↔ w~ ∈ Y )

∀f ∀g(f = g ↔ ∀~z.f (~z) = g(~z) The LUB Principle in Weyl’s System

Not valid: Every bounded set of real numbers has a sup. The reason: the following (illegal) abstract term, in which X : Set(Set(N4)) and y : Set(N4), uses a forbidden quantification:

SUP[X ] = {hn1, n2, n3, n4i | ∃y.y ∈ X ∧ hn1, n2, n3, n4i ∈ y}

Valid: Every bounded sequence of real numbers has a sup. The reason: the following (legal) term, in which f : N → Set(N4), defines the needed function SUP :(N → Set(N4)) → Set(N4):

SUP = λf .{hn1, n2, n3, n4i | ∃k.hn1, n2, n3, n4i ∈ f (k)} Topics Developed and Theorems Proved in Weyl’s System

Cauchy convergence criterion for sequences Heine-Borel theorem for coverings with sequences of intervals Infinite series and power series Elementary functions continuous functions The mean-value theorem Maxima/Minima of continuous functions on closed intervals Uniform continuity of continuous functions in closed intervals Differentiation and integration of continuous functions

Following Weyl, Feferman did a lot more in related predicative systems. He conjectured that all scientifically applicable mathematics can be developed in predicative systems. Reconstructing Weyl’s Work in Set Theory — why?

The basic notions of set theory are used in any branch and textbook of modern mathematics. Moreover, set theory is almost universally accepted as the foundational theory in which the whole of mathematics can (and should) be developed.

The typed systems of Weyl (and Feferman) include cumbersome duplications and unintuitive distinctions. Duplications in Weyl’s System

Let N = {n | n = n} While N is the type of the natural numbers, N is a term of type Set(N), denoting the set of these numbers. Hence N 6= N .

Let P = λX : Set(N){Y : Set(N) | ∀n.n ∈ Y → n ∈ X } While Set(N) is the type of sets of natural numbers, P(N ) is a term of type Set(Set(N)), denoting the set of sets of natural numbers. Hence Set(N) 6= P(N ).

Note that P(N ) is a set in Weyl’s system (but not a definite one). This is also a drawback from a predicativist point of view. (In contrast, P(P(N )) is not a set in Weyl’s system.) Our Framework for Axiomatic Set Theories

Extensionality:

∀z(z ∈ x ↔ z ∈ y) → x = y

The Comprehension Schema:

∀y(y ∈{ x | ϕ}↔ ϕ[y/x])

provided ϕ is safe w.r.t. x.

∈-induction:

(∀x(∀y(y ∈ x → ϕ[y/x]) → ϕ)) → ∀xϕ

Other Axioms: (AC, V=L, Martin, . . . ) Another Characterization of Impredicative Definitions

A definition is predicative if the class it defines is invariant under extension.

Poincare [1909, P. 463] writes: “Hence a distinction between two species of classifications, which are applicable to the elements of infinite collections: the predicative classifications, which cannot be disordered by the introduction of new elements; the non-predicative classifications, which are forced to remain without end by the introduction of new elements.” The Predicative Notion of Safety in Set Theory

Let Fv(ϕ) = {x1,..., xn, y1,..., ym}

ϕ is safe with respect to {x1,..., xn} if for every a1 ..., am, the value of (λ~y.{~x | ϕ})(~a) depends only on ~a, that is:

−→ n −→ −→ −→ n −→ −→ { x ∈ S2 | S2 |= ϕ( x , a )} = { x ∈ S1 | S1 |= ϕ( x , a )}

for every acceptable universes S1, S2 s. t. a1,..., an ∈ S1 ∩ S2.

ϕ is domain independent in the sense of database theory if ϕ Fv(ϕ).

ϕ is absolute in the sense of set theory if ϕ ∅. Reconstructing Weyl’s Work in Set Theory — How?

For Weyl safety was a property of a formula. However, without the guard of types, safety should become a relation between formulas and variables.

The notion of predicative safety is based on semantic intuition: universe independence. However, Weyl replaced it by syntactic approximation, defined in a static way. We shall do the same.

THE PROBLEM: What should be the syntactic properties of a predicative safety relation that would suffice for developing an adequate framework for predicative set theories? Safety Relations

Let L be a (first-order) language. A relation between formulas ϕ of L and subsets of Fv(ϕ) is a safety relation for L if it has the following properties of domain/universe independence:

If ϕ X and Y ⊆ X , then ϕ Y . ϕ ∅ if ϕ is atomic. ϕ {x} if ϕ ∈ {x 6= x, x = t, t = x}, and x 6∈ Fv(t). ¬ϕ ∅ if ϕ ∅. ϕ ∨ ψ X if ϕ X and ψ X . ϕ ∧ ψ X ∪ Y if ϕ X , ψ Y and Y ∩ Fv(ϕ) = ∅. ∃yϕ X − {y} if y ∈ X and ϕ X . Rudimentary Set Theory

We denote by RST the minimal safety relation allowed in our framework. It can inductively be defined as follows:

ϕ RST ∅ if ϕ is atomic.

ϕ RST {x} if ϕ ∈ {x = t, t = x, x 6= x, x ∈ t} (x 6∈ Fv(t)).

¬ϕ RST ∅ if ϕ RST ∅.

ϕ ∨ ψ RST X if ϕ RST X and ψ RST X .

ϕ ∧ ψ RST X ∪ Y if ϕ RST X , ψ RST Y and Y ∩ Fv(ϕ) = ∅.

∃yϕ RST X − {y} if y ∈ X and ϕ RST X .

RST (Rudimentary Set Theory) is the set theory which is induced in our framework by RST . RST : Some Examples

∅ =Df {x | x 6= x}.

{t1,..., tn} =Df {x | x = t1 ∨ ... ∨ x = tn}

ht, si =Df {{t}, {t, s}}.

{x ∈ t | ϕ} =Df {x | x ∈ t ∧ ϕ}, provided ϕ ∅.

{t(x) | x ∈ s} =Df {y | ∃x.x ∈ s ∧ y = t} S t =Df {x | ∃y.y ∈ t ∧ x ∈ y}

s × t =Df {x | ∃a∃b.a ∈ s ∧ b ∈ t ∧ x = ha, bi}

λx ∈ s.t =Df {hx, ti | x ∈ s} (where x 6∈ Fv(s)) S f (t)= Df {y | ∃z∃v(z ∈ f ∧ v ∈ z ∧ y ∈ v ∧ z = ht, yi)}

`RST a ∈ s → (λx ∈ s.t)(a) = t{a/x} Handling The Impredicative Comprehension Axioms

Each of the impredicative comprehension axioms of ZF can be captured (in a modular way) by adding to the definition of the safety relation a corresponding syntactic condition:

Separation: ϕ ∅ for every formula ϕ.

Powerset: x ⊆ t {x} if x 6∈ Fv(t). (Here ⊆ should better be taken as a new primitive.)

Replacement: ∃yϕ ∧ ∀y(ϕ → ψ) X provided ψ X , and X ∩ Fv(ϕ) = ∅. The system RST ω

Add to the language of RST a new constant ω (interpreted as the collection of natural numbers).

Let s(x) = x ∪ {x}. Add to the set of axioms of RST :

∀x(x ∈ ω ↔ ∀y ∈ s(x).y = ∅ ∨ ∃w ∈ s(x).y = s(w)) RST ω and J2

The minimal model of RST ω is J2.

Each a ∈ J2 is defined by some closed term of RST ω.

It can be shown that J2 (as a universe) and RST ω (as a theory) suffice for great parts (all?) of scientifically applicable mathematics.

However, this involves a lot of coding, as well as treating the collection of real numbers as a proper class. Introducing Iteration: The Use of Ancestral Logic (AL)

Languages in AL are defined like first-order languages with equality, but with the following additional clause: If ϕ is a formula, x, y are distinct variables which are free in ϕ, and s, t are terms, then( TCx,y ϕ)(s, t) is a formula.

The intended meaning of( TCx,y ϕ)(x, y) is:

ϕ(x, y)

∨∃w1.ϕ(x, w1) ∧ ϕ(w1, y)

∨∃w1∃w2.ϕ(x, w1) ∧ ϕ(w1, w2) ∧ ϕ(w2, y) ∨ ...

Unlike SOL, AL involves no new ontological commitments. Using AL in Our Framework

Safety Relations in AL are defined like in the case of FOL, but with the following additional condition (that respects d.i.):

(TCx,y ϕ)(x, y) X if ϕ X and {x, y} ∩ X 6= ∅

PZF is the minimal safety relation in AL.

PZF (Predicative Set Theory) is the set theory induced by PZF .

Expressive Power:

All finitary inductive definitions are available in PZF .

ω =Df {x | x = ∅ ∨ ∃y.y = ∅ ∧ (TCx,y x = y ∪{ y})(x, y)} Properties of PZF

The minimal model of PZF is Jωω = Lωω .

If t is a closed term of PZF then t defines an element of Jωω . Conversely, every element of Jωω is defined by some closed term of PZF .

In particular, J2, J3, ... , Jω, Jω2 , Jω3 , ... are so defined.

Jωω (as a universe) and PZF (as a theory) suffice for great parts (all?) of mainstream mathematics. This involves no coding, and the collection of real numbers can be treated as a set (e.g. as an element of Jω2 ). However, no such set can include all the “real numbers” which are definable in PZF and so are available in Jωω .