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From the Foundational Crisis of Mathematics to Explicit Mathematics

PhDs in Logic XI Gerhard J¨ ager

University of Bern Bern, April 2019

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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

• bjects of our perception or of our thought – which are called

elements of the set.

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Georg Cantor (1845 – 1918)

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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)

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Immediate reaction: Many forms of “restricted set theories”, avoiding Russell’s paradox. But central question remains: How “safe” are these restriced formalisms?

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Outline

1

Three main programmatic reactions

2

Over the last decades

3

Explicit mathematics

4

Universes

5

Mahloness and further up

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Three main programmatic reactions

Hilbert’s doctrine

Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben k¨

• nnen. (Nobody should be able to drive us out of

Paradise, the Cantor created us.)

David Hilbert (1862 – 1943)

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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 F0 that is justified by finite reasoning (some sort of finite combinatorics). (3) And then try to develop a sequence of increasing systems F0, F1, F2, . . . , Fk = F such that Fi establishes the consistency of Fi+1 by finite methods.

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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¨

• del’s results show that this program cannot work.
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Three main programmatic reactions

Brouwer’s dogma

Mathematics is an essentially languageless mental activity, based

• n a philosophy of mind and leading to a form of constructive

mathematics.

Luitzen Egbertus Jan Brouwer (1881 – 1966)

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Three main programmatic reactions

A non-constructive proof

Theorem

There are irrational numbers a and b such that ab is rational.

Proof.

We know from school that √ 2 is irrational. Now we distinguish the following two cases: (i) √ 2

√ 2 is rational. Then simply set a := b :=

√ 2. (ii) √ 2

√ 2 is irrational. Then we set a :=

√ 2

√ 2 and b :=

√ 2 and

• bserve:

ab = ( √ 2

√ 2) √ 2 =

√ 2

( √ 2· √ 2) =

√ 2

2 = 2.

This finishes the proof, but this argument does not tell us whether a is √ 2 or √ 2

√ 2.

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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”).

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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

• n the natural numbers (for arbitrary properties) do not require

foundational justification; further sets have to be introduced by purely predicative means.

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Three main programmatic reactions

Henri Poincar´ e (1854 – 1912) Hermann Weyl (1885 – 1955)

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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.

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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 Γ0.

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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 HAω. Systems of intuitionistic and constrictive set theory (IZF, CZF). Various type theories (MLTT, HoTT). and much more.

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Explicit mathematics

Explicit Mathematics

Solomon Feferman (1928 – 2016)

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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.

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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.

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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.

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Explicit mathematics

The element relation ∈ and the naming relation ℜ

t ∈ S :::

• bject t is an element of class S

(Strong form of polymorphism: an object may belong to many classes.) Equality of classes defined by S = T := ∀x(x ∈ S ↔ x ∈ T). Classes can be addressed via their names: ℜ(t, S) :::

• bject t is a name of class S.
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Explicit mathematics

Explicit representation and equality

(E1) ∃xℜ(x, S), (E2) ℜ(r, S) ∧ ℜ(r, T) → S = T, (E3) ℜ(r, S) ∧ S = T → ℜ(r, T). Some abbreviations: s ˙ ∈ t := ∃X(ℜ(t, X) ∧ s ∈ X), s ˙ = t := ∃X(ℜ(s, X) ∧ ℜ(t, X)), S ⊆ T := (∀x ∈ S)(x ∈ T), s ˙ ⊆ t := ∃X, Y (ℜ(s, X) ∧ ℜ(t, Y ) ∧ X ⊆ Y ), s ∈ ℜ := ∃Xℜ(s, X) (although ℜ is in general not a class).

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Explicit mathematics

Basic characteristics of this operational framework

Reconcile the intensional with the extensional point of view: Intensionality on the level of objects (names) and extensionality on the level of classes. The general universe of discourse simply is a patial combinatory algebra; typical examples: Kleene’s first and second model, the graph model, the (total) term model, . . . . Self-application of objects – we often call them operations – to each

• ther is possible; however, it does not necessarily produce a value.

The exact nature of these operations is purposely left open. The universe is open-ended but has some simple closure properties. No specific ideology.

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Explicit mathematics

The language L

Basic vocabulary: Variables for individuals: a, b, c, f , g, h, x, y, z, . . . . Variables for classes A, B, C, R, S, T, X, Y , Z, . . . . Many individual constants and a class constant N. Function symbol ◦ for (partial) term application. Relation symbols ↓, ∈, =, and ℜ. Indiividual terms (r,s,t,. . . ):

• ind. variables | ind. constants | (s ◦ t)

As usual: st := (s ◦ t) s1(s2 . . . sn) := s1s2 . . . sn := (. . . (s1s2) . . . sn).

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Explicit mathematics

Logic of partial terms (Beeson)

t↓ ::: term t has a value; s ≃ t := (s↓ ∨ t↓ → s = t).

Some characteristic properties

x↓. c↓ if c is a constant. st↓ → (s↓ ∧ t↓). A[t] → t↓ for atomic A[t]. A[t] ∧ t↓ → ∃xA[x].

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Explicit mathematics

Partial combinatory algebra

Combinatory axioms, pairing and projections

(PCA1) k = s. (PCA2) kab = a. (PCA3) sab↓ ∧ sabc ≃ (ac)(bc). (PCA4) p0a, b = a ∧ p1a, b = b, where a, b := pab.

Immediate consequences

λ-abstraction, fixed point theorem. A “computational engine”, acting on our universe.

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Explicit mathematics

A formula A is called stratified iff the relationsymbol ℜ does not occur in A; elementary iff it is stratified and does quantify over classes. Finite axiomatization of uniform elementary comprehension such that:

Theorem

For every elementary formula ϕ[u, v, W ] with at most the indicated free variables there exists a closed term tϕ such that:

1

• w ∈ ℜ → tϕ(

v, w) ∈ ℜ,

2 ℜ(

w, W ) → ∀x(x ˙ ∈ tϕ( v, w) ↔ ϕ[x, v, W ]). Hence, tϕ( v, w) is a name of {x : ϕ[x, v, W ])}. Comprehension for non-stratified formulas may lead to inconsisteny.

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Explicit mathematics

The natural numbers (N, 0, sN, pN, dN)

Some abbreviations: f : (Nk → N) := (∀x1, . . . , xk ∈ N)(f (x1, . . . xk) ∈ N)), t′ := sNt.

Basic N-axioms

(N1) 0 ∈ N ∧ a ∈ N → a′ ∈ N. (N2) a′ = 0 ∧ pN0 = 0 ∧ pN(a′) = a. (N3) x ∈ N ∧ y ∈ N ∧ x = y → dN(a, b, x, y) = a. (N4) x ∈ N ∧ y ∈ N ∧ x = y → dN(a, b, x, y) = b.

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Explicit mathematics

Class induction (C-IN)

0 ∈ S ∧ (∀x ∈ N)(x ∈ S → x′ ∈ S) → (∀x ∈ N)(x ∈ S).

Formula induction (L-IN)

ϕ[0] ∧ (∀x ∈ N)(ϕ[x] → ϕ[x′]) → (∀x ∈ N)ϕ[x]. The elementary theoy of classes EC is formulated in the classical logic of partial individual terms with equality.

Elementary theory of classes

EC := (E) + (PCA) + (N) + (el.comp.)

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Explicit mathematics

Theorem (First observation)

1 EC + (C-IN) ≡ ACA0 ≡ PA. 2 EC + (L-IN) ≡ ACA.

Remark (primitive recursion)

There exists a closed term rec such that EC + (C-IN) proves: a ∈ C ∧ n ∈ N ∧ (f : N × C → C) ∧ g = rec(a, f ) → (g : N → C) ∧ g(0) = a ∧ g(n′) = f (n, g(n)).

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Explicit mathematics

Adding join (J)

If a is a name and if f maps all elements of this class to names, then DJ[b, a, f ] says that b names the disjoint union of (fx : x ˙ ∈ a), b ˙ =

• x ˙

∈a

fx.

Join (J)

a ∈ ℜ ∧ (∀x ˙ ∈ a)(fx ∈ ℜ) → j(a, f ) ∈ ℜ ∧ DJ[ j(a, f ), a, f ].

Theorem

1 EC + (J) + (C-IN) ≡ ACA0 ≡ PA. 2 EC + (J) + (L-IN) ≡ Π0

1-CA<ε0 ≡ Σ1 1-AC.

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Explicit mathematics

Some ontological observations

Two forms of power classes

Strong power class (SP). For every class X there exists a class Y such that Y consists exactly of the names of all subclasses of X, ∀X∃Y ∀z(z ∈ Y ↔ ∃Z(ℜ(z, Z) ∧ Z ⊆ X)). Weak power class (WP). It only claims that for each class X there exists a class Y such that each element of Y names a subclass of X and for any subclass of X at least one of its names belongs to Y , ∀X∃Y ((∀z ∈ Y )(∃Z ⊆ X)(ℜ(z, Z)) ∧ (∀Z ⊆ X)(∃z ∈ Y )ℜ(z, Z)).

Remark

Even the uniform version of (WP) is consistent with EC.

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Explicit mathematics

Theorem

1 The names of a class never form a class, i.e.

EC ⊢ ∀X¬∃Y (Y = {z : ℜ(z, X)}).

2 Hence, (SP) is inconsistent with EC. 3 It is consistent with EC (though not provable there) to assume that

there exists the class of all names.

4 The theory EC + (J) proves that not all objects are names. 5 The theory EC + (J) proves the negation of (WP).

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Explicit mathematics

Class extensionality (Cl-Ext)

(∀x, y ∈ ℜ)(x ˙ = y → x = y). However, (Cl-Ext) is inconsistent with EC.

Operational extensionality (Op-Ext)

∀f , g( ∀x(fx ≃ gx) → f = g ). Then we have, for example, where (Tot) := ∀x, y(xy↓):

1 EC + (Op-Ext) + (Tot) is consistent. 2 EC + (Op-Ext) + ∀x(x ∈ N) is inconsistent. 3 EC + (Tot) + ∀x(x ∈ N) is inconsistent.

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Universes

Adding universes

A (predicative) universe is a class that is closed under elementary compre- hension and join and that consists of names only.

Universe

We write Univ[S] for the conjunction of the following formulas: (∀x ∈ S)(x ∈ ℜ). ∀ y(∀ z ∈ S)(tϕ( y, z) ∈ S), where tϕ is the comprehension term associated to the elementary frmula ϕ[x, y, Z]. ∀f (∀a ∈ S)((∀x ˙ ∈ a)(fx ∈ S) → j(a, f ) ∈ S). In addition, U[t] := ∃X(ℜ(t, X) ∧ Univ[X]).

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Universes

Universes can be regarded as the explicit analogies of regular sets or regular ordinals if the operations are interpreted as set-theoretic functions, admissible sets if the operations are interpreted as partial recursive functions.

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Universes

Basic ontological properties of universes

Universes do not contain their names

Univ[S] ∧ ℜ(a, S) → a / ∈ S. U[a] → a ˙ / ∈ a.

The names of a class cannot be in a single universe

Univ[S] → ∃x(ℜ(x, T) ∧ x / ∈ S).

EC + (J) does not prove the existence of universes

EC + (J) ⊢ ∃XUniv[X].

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Universes

The limit axioms (Lim)

The limit axiom (Lim). For a fresh constant ℓ:

(L1) a ∈ ℜ → ℓa ∈ ℜ, (L2) ℓa ∈ ℜ → U[ℓa] ∧ a ˙ ∈ ℓa.

Attention: Non-extensionality of ℓ

EC + (J) ⊢ (∃x, y ∈ ℜ)(x ˙ = y ∧ ℓx ˙ = ℓy).

Theorem (J¨ a)

1 |EC + (J) + (Lim)| + (C-IN) = Γ0 = ϕ(1, 0, 0). 2 |EC + (J) + (Lim)| + (L-IN) = ϕ(1, ε0, 0).

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Mahloness and further up

Explicit Mahlo

EC+(J)+(Lim) describes the explicit analogue of an inaccessible universe

• r of an recursively inaccessible universe. It is predicatively justified or

reducible according the the Feferman-Sch¨ utte approach. What if we go a step further? An ordinal α is callled a Mahlo ordinal iff (∀f : α → α)(∃β < α)(β ∈ Reg ∧ f : β → β). We live in a Mahlo world – roughly speaking – if for every class A and for every operation f that maps classes to classes there exists a universe U(A, F) such that A is represented in U(A, F) and f maps U(A, f ) to U(A, f ).

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Mahloness and further up

In the language of explicit mathematics: f ∈ (ℜ → ℜ) := (∀x ∈ ℜ)(fx ∈ ℜ), f ∈ (a → a) := (∀x ˙ ∈ a)(fx ˙ ∈ a).

Mahlo axioms (M)

(M1) a ∈ ℜ ∧ f ∈ (ℜ → ℜ) → m(a, f ) ∈ ℜ, (M2) m(a, f ) ∈ ℜ →

• U[m(a, f )] ∧ a ˙

∈ m(a, f ) ∧ f ∈ (m(a, f ) → m(a, f )).

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Mahloness and further up

About the predicative/impredicative status of (M)

These axioms have an impredicative character since they refer to total operations from ℜ to ℜ and introduce an element of ℜ. However, there is also a bottom up approach to (M): Assume that a is a name and f an operation from ℜ to ℜ. Now we (predicatively) form a universe that contains a and is closed under f . If this universe does already exist and m(a, f ) ∈ ℜ, then we are done; otherwise we add this universe as a new class and replace ℜ by ℜ ∪ {m(a, f )}.

Theorem (J¨ a, Strahm)

1 |EC + (J) + (M) + (C-IN)| = ϕ(ω, 0, 0). 2 |EC + (J) + (M) + (L-IN)| = ϕ(ε0, 0, 0).

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Mahloness and further up

Work in progress Higher reflection principles in explicit mathematics (long term). What are the explicit analogues of large cardinals? How to deal with classical and Bishop-style set theory in an explicit framework? Partially answered by Jaun in his recent PhD thesis. Univalence in explicit mathematics?

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Mahloness and further up

Thank you for your attention!

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