From the Foundational Crisis of Mathematics to Explicit Mathematics - PowerPoint PPT Presentation

From the Foundational Crisis of Mathematics to Explicit Mathematics PhDs in Logic XI Gerhard J¨ ager University of Bern Bern, April 2019 G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 1 / 44

How it all started: On a Thursday morning in 1895, Unter einer “Menge” verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer An- schauung oder unseres Denkens (welche die Elemente von M genannt werden) zu einem Ganzen. A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought – which are called elements of the set. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 2 / 44

Georg Cantor (1845 – 1918) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 3 / 44

To the foundational crisis of mathematics: The Russell set R := { x : x / ∈ x } However, then R ∈ R if and only if R / ∈ R , a contradiction! Bertrand Russell (1872 – 1970) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 4 / 44

Immediate reaction: Many forms of “restricted set theories”, avoiding Russell’s paradox. But central question remains: How “safe” are these restriced formalisms? G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 5 / 44

Outline Three main programmatic reactions 1 Over the last decades 2 Explicit mathematics 3 Universes 4 Mahloness and further up 5 G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 6 / 44

Three main programmatic reactions Hilbert’s doctrine Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben k¨ onnen. (Nobody should be able to drive us out of Paradise, the Cantor created us.) David Hilbert (1862 – 1943) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 7 / 44

Three main programmatic reactions The program of proof theory (Beweistheorie) The crucial steps (1) The eventual aim is a formal system F in which all of mathematics (or at least those parts relevant for us) can be formalized. (2) Start off from a basic system F 0 that is justified by finite reasoning (some sort of finite combinatorics). (3) And then try to develop a sequence of increasing systems F 0 , F 1 , F 2 , . . . , F k = F such that F i establishes the consistency of F i +1 by finite methods . G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 8 / 44

Three main programmatic reactions The underlying idea Formulas and proofs can be coded as finite sequences. Thus, by finite manipulations only, one can show that proofs of (0 = 1) cannot exist. However, G¨ odel’s results show that this program cannot work. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 9 / 44

Three main programmatic reactions Brouwer’s dogma Mathematics is an essentially languageless mental activity, based on a philosophy of mind and leading to a form of constructive mathematics. Luitzen Egbertus Jan Brouwer (1881 – 1966) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 10 / 44

Three main programmatic reactions A non-constructive proof Theorem There are irrational numbers a and b such that a b is rational. Proof. √ We know from school that 2 is irrational. Now we distinguish the following two cases: √ √ √ 2 is rational. Then simply set a := b := (i) 2 2. √ √ √ √ √ 2 is irrational. Then we set a := 2 and b := (ii) 2 2 2 and observe: √ √ √ √ √ √ √ 2) = 2 = 2 . a b = ( 2 ) 2 = ( 2 · 2 2 2 This finishes the proof, but this argument does not tell us whether a is √ √ √ 2 . 2 or 2 G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 11 / 44

Three main programmatic reactions Constructive formal systems Some characteristc properties of constructive systems Disjunction property: CS ⊢ A ∨ B ⇒ CS ⊢ A or CS ⊢ B . Existence property: CS ⊢ ∃ xA [ x ] ⇒ CS ⊢ A [ t ] for some term t . Constructive systems are based on intuitionistic logic (no “tertium non datur”). G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 12 / 44

Three main programmatic reactions The vicious circle principle (VCP) A definition of an object S is impredicative if it refers to a totality to which S belongs. A typical example: S = { n ∈ N : ( ∀ X ⊆ N ) ϕ [ X , n ] } ? : m ∈ S ( ∀ X ⊆ N ) ϕ [ X , m ] ϕ [ S , m ] m ∈ S . � � � Russell and Poincar´ e (around 1901 – 1906), later also Weyl VPC is the essential source of inconsistencies. The structure of the natural numbers and the principle of induction on the natural numbers (for arbitrary properties) do not require foundational justification; further sets have to be introduced by purely predicative means. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 13 / 44

Three main programmatic reactions Henri Poincar´ e (1854 – 1912) Hermann Weyl (1885 – 1955) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 14 / 44

Three main programmatic reactions Weyl’s foundational contributions Weyl showed, by way of examples, that many parts of mathematics can be developed in subsystems of second order arithmetic that are equiconsistent to Peano arithmentic PA. 1918: Das Kontinuum ; not much later, Weyl became a convert to Brouwerian intuitionistic constructivism. Independent of the classical versus intuitionistic question, Weyl always (from 1917 on) was critical of the Cantor-style set-theoretic foundations of mahematics: The set-theoretical foundations of mathematics are a house built to an essential extent on sand. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 15 / 44

Three main programmatic reactions In Das Kontinuum : ◮ The natural number system is a basic conception; proof and definition by induction are also basic. ◮ All other mathematical concepts (sets and functions) have to be introduced by explicit definitions. There are no completed totalities. ◮ Definitions which single out an object from a totality by reference to that totality are not permitted (Russell-Poincar´ e predicativity). ◮ Statements formulated in terms of these notions have a definite truth value (true or false). Around 1964/65: The limit of predicativity ` a la Feferman and Sch¨ utte, the ordinal Γ 0 . G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 16 / 44

Over the last decades A rich foundational landscape Classical mathematics Systems of set theory (ZFC, NBG), and their subsystems, e.g. KP. Subsystems of second order arithmetic and the program of Reverse Mathematics . Constructive mathematics Heyting arithmetic HA and Heyting arithmetic of higher type HA ω . Systems of intuitionistic and constrictive set theory (IZF, CZF). Various type theories (MLTT, HoTT). and much more. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 17 / 44

Explicit mathematics Explicit Mathematics Solomon Feferman (1928 – 2016) G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 18 / 44

Explicit mathematics Point of departure Systems of explicit mathematics introduced by S. Feferman in 1975. Since then they play an important role in foundational discussions: Original aim: formal framework for constructive mathematics, in particular Bishop-style constructive mathematics. First vesions of explicit mathematics based on intuitionistic logic; later formulated in a classical framework. Close relationship to systems of second order arithmetic and set theory; instrumental for reductive proof theory. Logical foundations of functional and object oriented programming languages. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 19 / 44

Explicit mathematics Feferman’s three classsic papers: A language and axioms for explicit mathematics, in: J. N. Crossley (ed.), Algebra and Logic, Lecture Notes in Mathematics 450, Springer, 1975; Recursion theory and set theory: a marriage of convenience, in: J. E. Fenstad, R. O. Gandy, G. E. Sacks (eds.), Generalized Recursion Theory II, Studies in Logic and the Foundations of Mathematics 94, Elsevier, 1978; Constructive theories of functions and classes, in: M. Boffa, D. van Dalen,K. McAloon (eds.). Logic Colloquium ’78, Studies in Logic and the Foundations of Mathematics 97, Elsevier, 1979. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 20 / 44

Explicit mathematics Basic ontology (modern approach) Formulated in language a language L with first and second order variables and constants. The general universe (first order objects) Unspecified general objects, (constructive) operations, bitstrings, programs, . . . . These objects form a partial combinatory algebra. Classes (second order objects) Classes are simply collections of objects. These classes help to “structure” the universe. As we will see, more versatile than “traditional” type theories. G. J¨ ager (Bern University) Fondational Crisis – Explicit Mathematics April 2019 21 / 44