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: 8z(z 2 x $ z 2 y) ! x = y The Comprehension Schema: 8y(y 2f x j 'g$ '[y=x]) Ideal, but inconsistent! Predicativity, Impredicativity, and Poincar´eVCP The inconsistency of the ideal calculus implies that not every formula ' can be used for constructing legitimate set terms of the form fx j 'g. 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 = fx 2 S j 9X :X 2 2 ^ A(X ) ^ x 2 X g 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) = fq 2 Q j 9X :X 2 2Q ^ X 2 A ^ q 2 X g 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 :fhx1;:::; xni j 'g 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 2 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 be a safe formula such that Fv[ ] ⊇ fx1;:::; xng, where xi : τi (1 ≤ i ≤ n). Then fhx1;:::; xni j g : Set(τ) 3 Let k > 0, and let t : Set(τ) be a term such that Fv[t] ⊇ fy1;:::; yk g, 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] ⊇ fy1;:::; yk g, 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.