cantor's naive set theory - uni-goettingen.de

From Hilbert to Gentzen and beyond

Reinhard Kahle

CMA & Departamento de MatematicaFCT, Universidade Nova de Lisboa

Hilbert Bernays Summer School 2016Gottingen

Partially funded by the FCT projects Hilbert’s 24th Problem, PTDC/MHC-FIL/2583/2014,and UID/MAT/00297/2013, and the Erasmus Docente scheme of the EU.

Hilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 1 / 75

Cantor’s Naive Set Theory

Cantor 1895

“By a set we understand every collection to a whole M of definite,well-differentiated objects m of our intuition or our thought.”

M = x |ϕ(x), m ∈ M ⇔ ϕ(m)


Russell’s Paradox

Russell 1901

R = x |x /∈ x

R?∈ R

R ∈ R ⇔ R ∈ x |x /∈ x⇔ R /∈ R

E

Historical Note: Zermelo found the same paradox independently here inGottingen:B. Rang and W. Thomas, Zermelo’s discovery of the ‘Russell Paradox’,Historia Mathematica 8(1), 1981, pp. 15–22.


Hilbert’s Concerns

M = x |ϕ(x), m ∈ M ⇔ ϕ(m)

Which ϕ(x) are allowed for meaningful (consistent) set formations?Cantor considered the paradoxes as reductio-ad-absurdum argumentsfor the non-existence of a set associated to the underlying “setformations”.Hilbert—as Frege—was not happy with this “a posteriori view”.

Hilbert ∼1905

Why is the totality of all sets not permissible?Why is the set of all real numbers a permissible collection?

Zermelo’s axiomatization appears to be one answer to Hilbert’squestions—but it doesn’t really answer “Why”!Other answers, notably by Poincare, Weyl, and Brouwer, restrict settheory so far, that certain “usual” mathematical arguments cannot beexecuted any longer, notably in Analysis.


Hilbert’s Programme

The paradoxes were one of the motivations for Hilbert’s FoundationalStudies (there are others which, however, we do not address here).

To secure mathematical reasoning, Hilbert proposed the followingstrategy for consistency proofs:

1 Formalize mathematical reasoning (proofs).2 Showing that no formalized proof can end in a false formula

(as, for instance, 0 = 1).

Apparently, this is a purely combinatorial question: proofs can berepresented by certain sequences of formulas, constructed by cleardefined rules, and all one would have to show is, that such a sequencecould never have a particular formula as last element.

Note

Hilbert is, by no means, a formalist who considers Mathematics as a gamewith formulas. Formal proofs are just representation of “normalmathematical proofs”.


Hilbert’s Programme

Initial “philosophical problem” (Poincare):the methods (in particular, induction) used in a “meta proof”(expressing that 0 = 1 never could be proven) are those which are atstake—thus, one runs in a vicious circle.

Solution (suggested to Hilbert by Brouwer in 1909):using a “weak” theory—whose consistency is beyond doubt—to provethe consistency of strong theories.


First-order languages

Definition

A first-order language L is a set of symbols which can be divided in thefollowing six (disjunctive) subsets:

logical symbols: ¬,∧,∨,→,∀,∃,=;constant symbols: C ⊆ ci |i ∈ N,function symbols: F ⊆ f j

i |i ∈ N, j ∈ N, j > 0,where f j

i is the i-th function symbol of arity j ;

relation symbols R ⊆ R ji |i ∈ N, j ∈ N,

where R ji is the i-th relation symbol of arity j ;

variables: x , y , z ,w , . . . , x0, x1, x2, . . . ;auxiliary signs:

”(“,

”)“,

”,“,

”.“.


First-order languages

According to the definition, for a concrete first-order language we haveonly to specify only the sets C, F , and R.

Examples

1 For the language LPA of the Peano arithmetic we have: C = 0,F = s,+, ·, and R = ∅, where s is a unary function symbol for thesuccessor function.

2 The language of set theory (without urelements) can be given byC = F = ∅ and R = ∈.


Terms

Definition

The terms of L are defined inductively as following:

1 Each variable is a term.

2 Each constant symbol is a term.

3 If t1, t2, . . . , tn are terms and f n is a n-ary function symbol (n > 0),then the expression f n(t1, t2, . . . , tn) is also a term.


Formulae

Definition

The formulae of L are defined inductively as follows:

1 If t1 and t2 are terms, then the expression t1 = t2 is a formula.

2 If t1, t2, . . . , tn are terms and Rn is a n-ary relation symbol (n ≥ 0),then the expression Rn(t1, t2, . . . , tn) is a formula.

3 If ϕ and ψ are formulae, then the following expressions are alsoformulae:

(¬ϕ), (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ→ ψ).

4 If ϕ is a formula and x a variable, then the expressions (∀x .ϕ) and(∃x .ϕ) are also formulae.


Hilbert-style calculus I

Definition

We define the Hilbert-style calculus H as a derivation system with thefollowing (logical) axioms and rules:

1 The following formulae are axioms:I ` ϕ→ (ψ → ϕ)I ` (ϕ→ (χ→ ψ))→ (ϕ→ χ)→ (ϕ→ ψ)I ` (¬ϕ→ ¬ψ)→ ψ → ϕI ` ϕ→ (ϕ ∨ ψ)I ` ψ → (ϕ ∨ ψ)I ` (ϕ→ χ)→ ((ψ → χ)→ (ϕ ∨ ψ → χ))I ` (ϕ ∧ ψ)→ ϕI ` (ϕ ∧ ψ)→ ψI ` ϕ→ (ψ → (ϕ ∧ ψ))


Hilbert-style calculus II

Definition2 Equality axioms.

I (u = u),I (u = w)→ (w = u),I (u1 = u2 ∧ u2 = u3)→ (u1 = u3),I (u1 = w1 ∧ · · · ∧ un = wn)→ (R(u1, . . . , un)→ R(w1, . . . ,wn)),I (u1 = w1 ∧ · · · ∧ um = wm)→ (t[u1, . . . , um] = t[w1, . . . ,wm]),

where u,w , u1, . . . are variables and constant symbols, R a n-aryrelation symbol, and t a term, in which u1, . . . , um or w1, . . . ,wm mayoccur.

3 Quantifier axioms:I ` (∀x .ϕ(x))→ ϕ(t)I ` ϕ(t)→ (∃x .ϕ(x))


Hilbert-style calculus III

Definition

As rules we have:

4 Modus Ponens.` ϕ→ ψ

` ϕ` ψ

5 Generalisation; let x be a variable not free in ϕ.

` ϕ→ ψ(x)

` ϕ→ ∀y .ψ(y)

` ψ(x)→ ϕ

` (∃y .ψ(y))→ ϕ


Proof in H

Definition

A proof of ϕ starting from a set of formulae Φ (in the Hilbert-stylecalculus H), is a finite sequence of formulae ψ1, ψ2, . . . , ψn with ψn = ϕ,and each of these formulae ψi is either

an axiom of H,

an element of Φ, or

is obtained from the previous formulae ψj , j < i , by an application ofa rule.

We say that ϕ is provable from Φ (in the Hilbert-style calculus H), andwrite Φ ` ϕ, if there exists a proof of ϕ starting from Φ.


Example

ϕ→ ϕ is not an axiom in our calculus.

Example

` (ϕ→ ((ϕ→ ϕ)→ ϕ))→ (ϕ→ (ϕ→ ϕ))→ (ϕ→ ϕ) Second axiom

` ϕ→ ((ϕ→ ϕ)→ ϕ) First axiom

` (ϕ→ (ϕ→ ϕ))→ (ϕ→ ϕ) Modus Ponens

` ϕ→ (ϕ→ ϕ) First axiom

` ϕ→ ϕ Modus Ponens


Peano arithmetic

We use the language of Peano arithmetic LPA = 0, s,+, ·.

Definition (Peano arithmetic)

Peano arithmetic PA comprises the following six non-logical axioms andthe following axiom scheme:

(PA1) ∀x .¬(s(x) = 0),

(PA2) ∀x , y .s(x) = s(y)→ x = y ,

(PA3) ∀x .x + 0 = x ,

(PA4) ∀x , y .x + s(y) = s(x + y),

(PA5) ∀x .x · 0 = 0,

(PA6) ∀x , y .x · s(y) = (x · y) + x .

The axiom scheme of complete induction:ϕ(0) ∧ (∀y .ϕ(y)→ ϕ(s(y)))→ ∀x .ϕ(x).

PA ` ϕ iff there is a finite set Φ of axioms of PA such that Φ ` ϕ.


Peano arithmetic

Hilbert’s Programme for PA: showing that PA 0 0 = 1.

Apparently unrelated question:

Is PA syntactically complete, i.e., does for every formula ϕ holds that:

PA ` ϕ or PA ` ¬ϕ ?

Godel’s First Incompleteness theorem shows that this is not the case.

Godel’s Second Incompleteness theorem shows that the FirstIncompleteness theorem entails the impossibility of a consistencyproof for PA (and all stronger systems) in the way Hilbert hadenvisaged them.


Godel’s First Incompleteness Theorem

The first incompleteness theorem shows that the Peano Arithmetic issyntactically incomplete. That means, there is a formula ϕ such that

PA 6` ϕ and PA 6` ¬ϕ.

The idea of the proof is quite simple. Consider the classical paradoxof the liar:

This sentence is false.

Obviously, the sentence can neither be true nor false withoutprovocating a contradiction.

In analogy, consider now the following Godel sentence:

This sentence is not provable.

If this sentence can be represented faithfully in the language ofPeano-Arithmetic, it can neither be provable nor refutable (i.e., itsnegation would be provable).


Two challenges

To formalize the Godel sentence “This sentence is not provable.” in PA wehave to solve two problems:

1 Formalizing provability.

2 Expressing the self-reference (“This sencence . . . ”).


The proof predicate

Formulas are strings of symbols, which can be coded by numbers, itsGodel number:

ϕ 7→ pϕq ∈ N.

Proofs are finite sequences of formulas (obeying the derivation rulesof the calculus); thus, a proof can be coded by a sequence of thecorresponding Godel numbers:

〈pϕ1q, pϕ2q, . . . , pϕnq〉 ∈ N.

All this coding can be done within the realm of primitive recursivefunctions.

With some technical work, one can define a primitive recursive relationBewPA such that BewPA(x , y) is true, if and only if x is the Godelnumber of a proof in PA of the formula with the Godel number y .


Representability

Let n is a term of the language of the formal theory T representing thenatural number n.

Definition

Let T be an arbitrary theory.

A relation R ⊆ Nn is numeralwise representable in T by a formula ϕ ifone has, for all natural numbers m1, . . . ,mn:

R(m1, . . . ,mn) is true if and only if T ` ϕ(m1, . . . , mn),We also say ϕ numerates the relation R in T .

ϕ binumerates R in T if it numerates it and one has also:R(m1, . . . ,mn) is false if and only if T ` ¬ϕ(m1, . . . , mn).


Representability

Theorem (Representation Theorem)

PA binumerates all primitive-recursive relations.

This theorem applies to BewPA and we have that there is a formulaBewPA in the language of PA with:

BewPA(m1,m2) is true if and only if PA ` BewPA(m1, m2)

BewPA(m1,m2) is false if and only if PA ` ¬BewPA(m1, m2).


A provability predicate

By definition of the relation BewPA we have for its representationBewPA in PA:

PA ` ϕ⇐⇒ PA ` BewPA(t, pϕq) for a closed term t

=⇒ PA ` ∃x .BewPA(x , pϕq)

⇐⇒ PA ` BPA(pϕq)

t is a sequence number of 〈pϕ0q, pϕ1q, . . . , pϕn−1q, pϕq〉.In short:

PA ` ϕ =⇒ PA ` BPA(pϕq) (1)

Note that we don’t have immediately the “missing” direction:

PA ` ∃x .BewPA(x , pϕq) =⇒ PA ` BewPA(t, pϕq)

In general, one cannot conclude from an existential statement like∃x .BewPA(x , pϕq) that there is also a closed term which exemplifiessuch an x .


Diagonalization lemma

Theorem (Diagonalization lemma)

Let ϕ(x) be a formula with exactly one free variable x . Then there is asentence ψ such that:

PA ` ψ ↔ ϕ(pψq).

Proof.

Define ϑ(x) as ϕ(Sub(x ,Num(x))). Let m be pϑ(x)q and let ψ be ϑ(m).

ψ ↔ ϑ(m)

↔ ϕ(Sub(m,Num(m)))

↔ ϕ(Sub(pϑ(x)q, pmq))

↔ ϕ(pϑ(m)q)

↔ ϕ(pψq)

ψ expresses “I have the property ϕ”.Hilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 24 / 75

Godel’s First Incompleteness Theorem

Theorem (First Incompleteness Theorem; Godel 1931)

Assume, PA is consistent. Then, there is a sentence ϕ such that:

1 PA 6 ` ϕ;

2 If PA ` BPA(pϕq)⇒ PA ` ϕ, then PA 6 ` ¬ϕ.

Proof.

According to the diagonalization lemma, there is a sentence ϕ such that

PA ` ϕ↔ ¬BPA(pϕq). (∗)1 Assume PA ` ϕ. With (1) we have PA ` BPA(pϕq). With (∗) it

follows PA ` ¬BPA(pϕq) in contradiction to the consistency of PA.

2 Assume PA ` ¬ϕ. With (∗) we have PA ` ¬¬BPA(pϕq) and alsoPA ` BPA(pϕq). Because of the additional premise this gives PA ` ϕ,again in contradiction to the consistency of PA.


First Incompleteness Theorem: generic form

The premise PA ` BPA(pϕq)⇒ PA ` ϕ in the second casecorresponds to the ω-consistency which was assumed by Godel in hisoriginal paper.

In 1936, B. J. Rosser found a trick to avoid this condition, using amodified proof predicate BewR “on top” of Godel’s proof.

The result can be extended to any consistent, recursive extension ofPA:

Theorem (First Incompleteness Theorem)

Assume, that T is a consistent, recursive extension of PA. Then, there is asentence ϕ such that:

1 T 6 ` ϕ;

2 T 6 ` ¬ϕ.


Godel’s second incompleteness theorem

Godel’s second incompleteness theorem says that a theory, which hasat least the expressive power of Peano Arithmetic, cannot prove itsown consistency.

Using the techniques developed so far, consistency of a theory T canbe easily expressed as:

ConT ⇐⇒ ¬BT (pΛq)

where Λ is an arbitrary contradictory (false) formula, for instance,0 = s(0).

We say that a theory does not prove it own consistency if we have:

T 6 ` ConT .


The idea of the proof of Godel II

First we consider, again, only PA.In a sloppy formulation, the idea for the proof of the secondincompleteness theorem is to formalize the proof of the firstincompleteness theorem in PA.

1 If PA 6` ϕ, PA is obviously consistent (as an inconsistent theory provesevery formula). Thus:

PA 6` ϕ =⇒ PA is consistent.

2 The first incompleteness theorem states, for the chosen ϕ:

PA is consistent =⇒ PA 6` ϕ.

The formalization of both arguments within PA will show that this ϕis equivalent to the consistency statement of PA:

PA ` ¬BPA(pϕq)↔ ConPA

PA ` ϕ↔ ConPA.Hilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 28 / 75

Provability conditions

For the proof of the first incompleteness theorem we used thefollowing property of B:

PA ` ϕ =⇒ PA ` BPA(pϕq) (1)

For the proof of the second incompleteness theorem, we need the twoadditional properties of BPA:

PA ` BPA(pϕq)→ BPA(pBPA(pϕq)q) (2)

PA ` [BPA(pϕq) ∧ BPA(pϕ→ ψq)]→ BPA(pψq) (3)

(2) and (3) do not follow any longer directly from the representabilitytheorem. But they can be proven for BPA (with some hard work).

The three conditions are called Hilbert-Bernays-Lob derivablityconditions. They can be studied independently, and in an abstractform they are the base of provability logic.


Godel’s second incompleteness theorem

Theorem (Second incompleteness theorem)

Assume PA is consistent. Then we have:

PA 6 ` ConPA.


Proof of Godel’s second incompleteness theorem

Let ϕ be such that: PA ` ϕ ↔ ¬BPA(pϕq) (?)

PA ` Λ → ϕ Ex-falso-quodlibet

PA ` BPA(pΛq) → BPA(pϕq) (1) and (3)

PA ` ¬BPA(pϕq) → ¬BPA(pΛq) Contrapositive

PA ` ϕ → ¬BPA(pϕq) (?)

PA ` ϕ → ¬BPA(pΛq) Logical reasoning

PA ` ϕ → ConPA Definition of ConPA

PA ` BPA(pϕq) → ¬ϕ Contrapositive of (?)

PA ` BPA(pBPA(pϕq)q) → BPA(p¬ϕq) (1) and (3)

PA ` BPA(pϕq) → BPA(pBPA(pϕq)q) (2)

PA ` BPA(pϕq) → BPA(p¬ϕq) Logical reasoning

PA ` BPA(pϕq) → BPA(pϕ ∧ ¬ϕq) (1), (3) and logical reasoning

PA ` BPA(pϕq) → BPA(pΛq) Definition of Λ

PA ` ¬BPA(pΛq) → ¬BPA(pϕq) Contrapositive

PA ` ConPA → ϕ Definition of ConPA and (?)

As PA 6 ` ϕ we have also PA 6 ` ConPA.Hilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 31 / 75

Godel’s second incompleteness theorem; generic version

Theorem (Second incompleteness theorem; Godel 1931)

Assume, that T is a consistent, recursive extension of PA. Then

T 6 ` ConT .


Why reasoning in PA about PA?

Assume, the second incompleteness theorem would not hold, and itwould be the case that PA ` ConPA.

Obviously, such a proof would not give any evidence for theconsistency of PA: if PA would be incosistent, every formula would beprovable, in particular also ConPA.

The significance of the second incompleteness theorem (as given here)is based on an immediate corollary: if PA cannot prove its consistency,no weaker theory—in particular, any subsystem of PA—could do so.

But this was the idea in Hilbert’s programme: using finitisticmathematics—which is is supposed to be a subsystem of PA—toprove the consistency of PA (and other theories).


Consistency Proofs after Godel

For PA, we may consider the following three alternative approaches(all of them already discussed by Godel as early as 1938):

1 Intuitionistic Arithmetic: double negation interpretation. (Kolmogorov1925; Godel 1933; Gentzen 1936)

2 Primitive-recursive arithmetic with transfinite induction up to theordinal ε0 (Gentzen 1936)

3 Functionals of higher type: Godel’s T ; Dialectica interpretation (Godel1958)

What about stronger systems, first of all Analysis?

In the following we will pursue a little bit further Ordinal Analysis inGentzen-style proof theory.The following slides are taken with permission from a course given by

Michael Rathjen in 2005.


Sequent Calculus

A sequent is an expression Γ ⇒ ∆ where Γ and ∆ are finitesequences of formulae A1, . . . ,An and B1, . . . ,Bm, respectively.Σ ⇒ ∆ is read, informally, as Γ yields ∆ or, rather, theconjunction of the Ai yields the disjunction of the Bj .

In particular,

• If Γ is empty, the sequent asserts the disjunction of the Bj .

• If ∆ is empty, it asserts the negation of the conjunction of theAi .

• if Γ and ∆ are both empty, it asserts the impossible, i.e. acontradiction.

We use upper case Greek letters Γ,∆,Λ,Θ,Ξ . . . to range overfinite sequences of formulae.


Sequent Calculus

Identity AxiomA ⇒ A

where A is any formula. In point of fact, one could limit this axiomto the case of atomic formulae A.

CUTΓ ⇒ ∆,A A,Λ ⇒ Θ

CutΓ,Λ ⇒ ∆,Θ

A is called the cut formula of the inference.


Sequent Calculus

Structural Rules Exchange, Weakening, Contraction

Γ,A,B,Λ ⇒ ∆ XlΓ,B,A,Λ ⇒ ∆

Γ ⇒ ∆,A,B,Λ XrΓ ⇒ ∆,B,A,Λ

Γ ⇒ ∆ WlΓ,A ⇒ ∆

Γ ⇒ ∆ WrΓ ⇒ ∆,A

Γ,A,A ⇒ ∆ ClΓ,A ⇒ ∆

Γ ⇒ ∆,A,A CrΓ ⇒ ∆,A


Sequent Calculus

LOGICAL INFERENCES

Negation

Γ ⇒ ∆,A¬ L¬A, Γ ⇒ ∆

B, Γ ⇒ ∆¬R

Γ ⇒ ∆,¬B

Implication

Γ ⇒ ∆,A B,Λ ⇒ Θ→ L

A → B, Γ,Λ ⇒ ∆,Θ

A, Γ ⇒ ∆,B→ R

Γ ⇒ ∆,A → B


Sequent Calculus

Conjunction

A, Γ ⇒ ∆∧ L1

A ∧ B, Γ ⇒ ∆

B, Γ ⇒ ∆∧ L2

A ∧ B, Γ ⇒ ∆

Γ ⇒ ∆,A Γ ⇒ ∆,B∧R

Γ ⇒ ∆,A ∧ B

Disjunction

A, Γ ⇒ ∆ B, Γ ⇒ ∆∨ L

A ∨ B, Γ ⇒ ∆

Γ ⇒ ∆,A∨R1

Γ ⇒ ∆,A ∨ B

Γ ⇒ ∆,B∨R2

Γ ⇒ ∆,A ∨ BHilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 39 / 75

Sequent Calculus

Quantifiers

F (t), Γ ⇒ ∆∀ L∀x F (x), Γ ⇒ ∆

Γ ⇒ ∆,F (a)∀R

Γ ⇒ ∆,∀x F (x)

F (a), Γ ⇒ ∆∃ L∃x F (x), Γ ⇒ ∆

Γ ⇒ ∆,F (t)∃R

Γ ⇒ ∆, ∃x F (x)

In ∀L and ∃R, t is an arbitrary term. The variable a in ∀R and ∃Lis an eigenvariable of the respective inference, i.e. a is not to occurin the lower sequent.


Sequent Calculus

The formulae in a logical inference marked blue are called theminor formulae of that inference, while the red formula is theprincipal formula of that inference. The other formulae of aninference are called side formulae.

A proof (aka deduction or derivation) D is a tree of sequentssatisfying the following conditions:

• The topmost sequents of D are identity axioms.

• Every sequent in D except the lowest one is an upper sequentof an inference whose lower sequent is also in D.


Sequent Calculus

The INTUITIONISTIC case

The intuitionistic sequent calculus is obtained by requiring that allsequents be intuitionistic. A sequent Γ ⇒ ∆ is said to beintuitionistic if ∆ consists of at most one formula.

Specifically, in the intuitionistic sequent calculus there are noinferences corresponding to contraction right or exchange right.


Sequent Calculus

Our first example is a deduction of the law of excluded middle.

A ⇒ A ¬R⇒ A,¬A ∨R⇒ A, A ∨ ¬A Xr⇒ A ∨ ¬A, A ∨R⇒ A ∨ ¬A, A ∨ ¬A Cr⇒ A ∨ ¬ANotice that the above proof is not intuitionistic since it involvessequents that are not intuitionistic.


Sequent Calculus

The second example is an intuitionistic deduction.

F (a) ⇒ F (a)∃R

F (a) ⇒ ∃x F (x)¬L¬∃x F (x),F (a) ⇒ Xl

F (a), ¬∃x F (x) ⇒¬L¬∃xF (x) ⇒ ¬F (a)∀R¬∃x F (x) ⇒ ∀x ¬F (x)→R⇒ ¬∃x F (x) → ∀x ¬F (x)


Cut Elimination

Cut Elimination (Gentzen’s Hauptsatz)

If a sequent Γ ⇒ ∆ is provable,

then it is provable without cuts.

Here is an example of how to eliminate cuts of a special form:

A, Γ ⇒ ∆,B →RΓ ⇒ ∆,A → B

Λ ⇒ Θ,A B,Ξ ⇒ Φ →LA → B,Λ,Ξ ⇒ Θ,Φ

CutΓ,Λ,Ξ ⇒ ∆,Θ,Φ

is replaced by

Λ ⇒ Θ,A A, Γ ⇒ ∆,BCut

Λ, Γ ⇒ Θ,∆,B B,Ξ ⇒ ΦCut

Γ,Λ,Ξ ⇒ ∆,Θ,Φ


Cut Elimination

Remarks

• The proof of the cut elimination theorem is rather intricate asthe process of removing cuts interferes with contraction.

The possibility of contraction accounts for the high cost ofeliminating cuts. Let |D| be the height of the deduction D.Also, let rank(D) be supremum of the lengths of cut formulaeoccurring in D. Turning D into a cut-free deduction of thesame end sequent results, in the worst case, in a deduction ofheight

H(rank(D), |D|)where

H(0, n) = n H(k + 1, n) = 4H(k,n).


Cut Elimination

• Cut-free proofs aren’t suitable for the mathematical practice.The cut formulae in a proof usually carry the idea of the proof(lemmata). Removing cuts not only makes proofs longer butalso renders them less understandable.


Cut Elimination

The Hauptsatz has an important corollary.

The Subformula Property

If a sequent Γ ⇒ ∆ is provable, then it has a deductionall of whose formulae are subformulae of the formulae ofΓ and ∆.

Corollary

A contradiction, i.e. the empty sequent, is not deducible.


Mathematical Theories

While mathematics is based on logic, it cannot be developed solelyon the basis of pure logic. What is needed in addition are axiomsthat assert the existence of mathematical objects and theirproperties. Logic plus axioms gives rise to (formal) theories such asPeano arithmetic or the axioms of Zermelo-Fraenkel set theory.



What happens when we try to apply the procedure of cutelimination to theories? Well, axioms are poisonous to thisprocedure. It breaks down because the symmetry of the sequentcalculus is lost. In general, we cannot remove cuts from deductionsin a theory T when the cut formula is an axiom of T . However,sometimes the axioms of a theory are of bounded syntacticcomplexity. Then the procedure applies partially in that one canremove all cuts that exceed the complexity of the axioms of T .



This gives rise to

partial cut elimination.

This is a very important tool in proof theory. For example, it worksvery well if the axioms of a theory can be presented as atomicintuitionistic sequents (also called Horn clauses), yielding thecompleteness of Robinsons resolution method.



Partial cut elimination also pays off in the case of fragments of PAand set theory with restricted induction schemes, be it inductionon natural numbers or sets. This method can be used to extractbounds from proofs of Π0

2 statements in such fragments.


Peano Arithmetic (idea – Takeuti)

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Peano Arithmetic

Gentzen’s resultGerhard Gentzen showed that transfinite induction up to the

ordinal

ε0 = supω, ωω, ωωω

, . . . = least α. ωα = α

suffices to prove the consistency of Peano Arithmetic, PA. Toappreciate Gentzen’s result it is pivotal to note that he appliedtransfinite induction up to ε0 solely to primitive recursivepredicates and besides that his proof used only finitisticallyjustified means.


Peano Arithmetic

Hence, a more precise rendering of Gentzen’s result is

F+ PR-TI(ε0) ` Con(PA), (1)

where F signifies a theory that is acceptable in finitism (e.g.F = PRA = Primitive Recursive Arithmetic) and PR-TI(ε0) standsfor transfinite induction up to ε0 for primitive recursive predicates.Gentzen also showed that his result is best possible in that PAproves transfinite induction up to α for arithmetic predicates forany α < ε0.


Peano Arithmetic

The compelling picture conjured up by the above is that thenon-finitist part of PA is encapsulated in PR-TI(ε0) and therefore“measured” by ε0, thereby tempting one to adopt the followingdefinition of proof-theoretic ordinal of a theory T :

|T |Con = least α. PRA+ PR-TI(α) ` Con(T ). (2)


Peano Arithmetic (with ω rule)

Full arithmetic, i.e. PA, does not even allow for partial cutelimination since the induction axioms have unbounded complexity.

However, one can remove the obstacle against cut elimination in adrastic way by going infinite. The so-called ω-rule consists ofthe two types of infinitary inferences:

Γ ⇒ ∆,F (0); Γ ⇒ ∆,F (1); . . . ; Γ ⇒ ∆,F (n); . . .ωR

Γ ⇒ ∆, ∀x F (x)

F (0), Γ ⇒ ∆; F (1), Γ ⇒ ∆; . . . ;F (n), Γ ⇒ ∆; . . .ωL∃x F (x), Γ ⇒ ∆

The price to pay will be that deductions become infinite objects,i.e. infinite well-founded trees.



The sequent-style version of Peano arithmetic with the ω-rule willbe termed PAω.

PAω has no use for free variables. Thus free variables are discardedand thus all terms are closed. All formulae of this system aretherefore closed, too.The numerals are the terms n, where 0 = 0 and n + 1 = Sn. Weshall identify n with the natural number n. All terms t of PAω

evaluate to a numeral n.



PAω has all the inference rules of the sequent calculus except for∀R and ∃L. In their stead, PAω has the ωR and ωL inferences.

The Axioms of PAω are the following:

•⇒ A

if A is a true atomic sentence.

•B ⇒

if B is a false atomic sentence.

•F (s1, . . . , sn) ⇒ F (t1, . . . , tn)

if F (s1, . . . , sn) is an atomic sentence and si and ti evaluateto the same numeral.



With the aid of the ω-rule, induction becomes logically deduciblein infinitary logic.

Theorem For every n there is a finite deduction Dn of the sequent

F (0),∀x [F (x) → F (Sx)] ⇒ F (n).

Proof. Since B, Γ ⇒ B is deducible for every formula B andsequence Γ, we obtain D0.

Let ∆ := F (0), ∀x [F (x) → F (Sx)]. From Dn we obtain Dn+1 asfollows:

Dn

∆ ⇒ F (n).D∗

F (Sn),∆ ⇒ F (Sn)→ L

F (n) → F (Sn),∆ ⇒ F (S(n))∀L∀x [F (x) → F (Sx)],∆ ⇒ F (S(n))Struc

F (0), ∀x [F (x) → F (Sx)] ⇒ F (S(n))Hilbert Bernays Summer School 2016 From Hilbert to Gentzen and beyond 64 / 75


A final application of ωR yields an infinite deduction

D0

∆ ⇒ F (0). . .

Dn

∆ ⇒ F (n). . .

ωRF (0),∀x [F (x) → F (Sx)] ⇒ ∀x F (x)

∧LF (0) ∧ ∀x [F (x) → F (Sx)], ∀x [F (x) → F (Sx)] ⇒ ∀x F (x) Xl∀x [F (x) → F (Sx)], F (0) ∧ ∀x [F (x) → F (Sx)] ⇒ ∀x F (x)

∧LF (0) ∧ ∀x [F (x) → F (Sx)], same ⇒ ∀x F (x) Cl

F (0) ∧ ∀x [F (x) → F (Sx)] ⇒ ∀x F (x)→R⇒ F (0) ∧ ∀x [F (x) → F (Sx)] → ∀x F (x)



Cut Elimination for PAω

We want to measure the height and cut rank of a PAω

deduction D.We will notate this by

D α

kΓ ⇒ ∆ .

The above relation is defined inductively following the buildup ofthe deduction D.

For the cut rank we need the definition of the length, |A| of aformula:

• |A| = 0 if A is atomic;

• |¬A0| = |A0|+ 1;

• |A02A1| = max(|A0,A1|) + 1 where 2 = ∧,∨,→;

• |∃x F (x)| = |∀x F (x)| = |F (0)|+ 1.



Suppose the last inference of D is of the form

D0

Γ0 ⇒ ∆0. . .

Dn

Γn ⇒ ∆n. . . n < τ

IΓ ⇒ ∆

where τ = 1, 2, ω and the Dn are the immediate subdeductions ofD. If

Dnαn

kΓn ⇒ ∆n

and αn < α for all n < τ then

D α

kΓ ⇒ ∆

providing that in the case of I being a cut with cut formula A wealso have |A| < k.



We also just write PAωα

kΓ ⇒ ∆ if there exists a PAω deduction

D α

kΓ ⇒ ∆ .



Embedding Theorem If PA ` Γ ⇒ ∆ then

PAωω+m

kΓ ⇒ ∆

for some m, k < ω.



Reduction Lemma If PAωα

kΓ ⇒ ∆,A and PAω

β

kA,Λ ⇒ Θ

with k = |A|, then

PAωα#β

kΓ,Λ ⇒ ∆,Θ .



Theorem If PAωα

k+1Γ ⇒ ∆, then PAω

ωα

kΓ ⇒ ∆.

Cut Elimination Theorem If PAωα

n Γ ⇒ ∆, then

PAωωω..

.ωα

0Γ ⇒ ∆ ωω..

.ωα

︸︷︷︸n times

.


Ordinal Analysis

The compelling picture conjured up by the above is that thenon-finitist part of PA is encapsulated in PR-TI(ε0) and therefore“measured” by ε0, thereby tempting one to adopt the followingdefinition of proof-theoretic ordinal of a theory T :

|T |Con = least α. PRA+ PR-TI(α) ` Con(T ). (2)

|PA|Con = ε0.

So far, so good — but what does it mean?

And how the story goes on?


Gentzen’s Theorem

On the “positive side”, we have obtained a consistency proof whichfits, at least in part, Hilbert’s orginial aims.

I Why?I Kreisel (1979) reports of “[a] familiar joke . . . of Weyl who was

astonished that one should use ε0-induction to prove the consistency ofordinary, that is ω-induction”.

I For sure, Hermann Weyl considered this remark only as a joke, as hewill have known exactly what’s going on here: the induction up to ε0 isapplied to quantifier-free formulas, while the induction of PA applies toarbitrary formulae.

I And universal quantification — which was eliminated by Gentzen in theinduction schemata — was at the very bottom of Hilbert’s concerns,much more than, for instance, the tertium-non-datur.

On the “negative side” one has to emphasise, that the consistency ofPA is not at issue at all.


Gentzen’s Theorem

“Much nonsense has been pronounced about Gentzen’s work, even byextremely distinguished people. Consistency is not really the mainissue at all. He did reveal fine structure in the unprovability ofconsistency of PA, as a consequence of much deeper generalmethodology.” Angus Macintyre, 2005.

And we have now an exact measure for the “transfinite amount” ofPA, i.e., its proof-theoretic ordinal ε0.


Ordinal Analysis of stronger systems

The natural question is now, what is the proof-theoretic ordinal ofAnalysis?

Second-Order Arithmetic (aka Analysis)

We consider a second-order language, which allows, next to naturalnumbers, for set variables. In addition to some version of the Peanoaxioms we include full comprehension:

∃X∀x .x ∈ X ↔ ϕ(x),

for each formula ϕ where X does not occur.

This comprehension scheme is highly impredicative:I ϕ(x) may contain a universal set quantification ∀Y . . . . . And this

quantifier is supposed to include the set X we are just introducing bythis comprehension. Thus, the definition of X depends on itself!



As a matter of fact, an ordinal analysis of second-order arithmetic isstill out of reach.

But one can treat restricted forms of comprehension, i.e.,second-order arithmetic where the comprehension axiom is restrictedto formulae ϕ(x) of restricted syntactical complexity.

ACA0 is a theory for arithmetical comprehension:I Comprehension is restricted to arithmetical formulas, which are

formulas without set quantifiers (but which may contain free setvariables).

I The index 0 indicates that we have induction only for sets (not forarbitrary formulas).

I ACA0 is a conservative extension of Peano Arithmetic (and, therefore,has the same proof-theoretic strength).

Π11-CA0 restricts comprehension to Π1

1 formulas, i.e., formulas of theform ∀X .ϕ(X ), with ϕ(X ) set-quantifier free.


Simpson on Π11-CA0

16

Π11-CA0 obviously includes ACA0. The proof-theoretic ordinal of Π1

1-CA0

is too complicated to describe here, but we briefly explain why. Suppose wewanted to replace the Π1

1 comprehension axiom the same way we replacedthe arithmetic comprehension axiom. The main case is where we introducesome formula on the left-hand side:

ΓACA0 ,Γ, ∀x(x ∈ X ↔ ∀Y φ(x, Y ))⇒ Σ.

In ACA0, we could replace the variable X with ∀Y φ(x, Y ) everywhere itappeared in the deduction. Here, if we can try to do the same, we discoverthat the variable X could have been used to deduce instances of ∃Y ¬φ(x, Y )!In other words, the set X consisting of those x where ∀Y φ(x, Y ) was truemight have used the set X itself to demonstrate that ∀Y φ(x, y) was false forsome particular x!

This is not a minor issue. Π11-CA0 introduces a deep new obstacle to

cut-elimination: which elements belong to a set defined by a Π11 formula

depends on a quantification over all sets, including the set which is in theprocess of being defined. This is a genuine circularity, and raises genuinenew difficulties, both mathematically and philosophically. (We should con-sider that it is not obvious, when stated like this, that Π1

1 comprehension isactually well-defined at all: how can we be sure that we won’t accidentallywrite down some set X with the property that x ∈ X iff x 6∈ X?) This the-ory is impredicative: the meaning of a set defined by a Π1

1 formula dependsessentially on a family of objects—the sets of numbers—which has not yetbeen completely constructed.

For the sake of comparison with the literature, we mention that the prooftheoretic ordinal of Π1

1-CA0 is given as the limit of a sequence of ordinals,with most (though not all) of the technical work needed to produce the firstone (the later ones iterate the same idea relative to the first one). The firstof these ordinals is known as the Howard-Bachmann ordinal, and is usuallywritten ψεΩ+1. Here Ω is (a name for) some very large ordinal, larger thanany other in the system—for example, ℵ1—εΩ+1 is the first fixed point ofthe α 7→ ωα which is larger than Ω, and ψ is a collapsing function whichtakes a very large ordinal and “collapses” it to a smaller one. (Alternatively,one can view εΩ+1 as representing the function α 7→ εα+1 and ψ as beingan operation which chooses an ordinal which is both a fixed point of thisfunction and also which is larger than the chosen fixed point of all “easilydefined” functions which grow more slowly.) Note that this definition isitself impredicative—we define the ordinal in terms of larger ordinals (orthe class of functions on ordinals).


Simpson on Π11-CA0

16


1-CA0



ΓACA0 ,Γ, ∀x(x ∈ X ↔ ∀Y φ(x, Y ))⇒ Σ.










Simpson on Π11-CA0

16


1-CA0



ΓACA0 ,Γ, ∀x(x ∈ X ↔ ∀Y φ(x, Y ))⇒ Σ.











The proof-theoretic analysis of subsystems of second-order Analysis isoften carried out by use of (proof-theoretically equivalent) theories inother frameworks, as

I Kripke-Platek set theoryI Feferman’s systems of Explicit MathematicsI Martin-Lof Type Theory

Thus, ordinal analysis covers actually a wide range of differentmathematical frameworks.



The ordinal analysis of impredicative systems (as Π11-CA0 or Π1

2-CA0)requires rather involved techniques, already to obtain notationsystems for the ordinals.

There are skeptical views on this:“Even if one succeeds in reducing the system (Π1

2-CA)± BI to aconstructive system (whether evidently so or not), one can hardlyexpect that doing so will appreciably increase one’s belief in itsconsistency (if one has any doubts about that in the first place) inview of the difficulty of checking the extremely complicated technicalwork needed for its ordinal analysis.” Feferman, 2000.

And one may agree with Hao Wang:“But what is clear from G[odel]’s work is the implausibility ofHilbert’s idea that somehow the foundations for mathematics can besecured once and for all by consistency proofs using certainparticularly transparent methods.” Hao Wang, 1987.

But . . .



But: as for Gentzen’s consistency proof of PA, the ordinal analysis ofsubsystems of second-order Arithmetic “reveals fine structure” of thetheories under considerations which can be (mathematically)rewarding.

It also gives us a linear measure for the strength of theories.

The used tools are of interest by themselves, alongside thecomplementary tools obtained by Godelization and functionalinterpretation.


A note on two different proof-theoretic tools

Godelization: ϕ 7→ pϕq.

Cut elimination (etc.): D D′


Re-evaluating Hilbert

Hilbert 1926

Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreibenkonnen.

No one shall be able to expel us from the paradise that Cantor has createdfor us!

This is the leitmotiv of Hilbert’s proof-theoretic programme.

If Godel shows that finitism is not enough to ensure this, we have togive up the philosophical base rather than parts of Mathematics.

One could also say that it wasn’t “Hilbert’s Programme” which failed,but the underlying philosophy.

And even if Gentzen’s extension of the “finitistic standpoint” is notyet enough, we simply have to look out for another philosophy. . .


Re-evaluating Hilbert

But if you insist in a “failure” of Hilbert’s Programme, let us draw ona comparison here:

I Nobody will deny that Columbus failed to find the sea route to India;I but he didn’t sink in the Ocean,I he discovered America.

I In the same way, Hilbert’s original programme didn’t succeed;I but it didn’t sink in inconsistency,I it discovered a rich hierarchy of stronger and stronger mathematical

theories.

And it was Gentzen who provided with the tools to explore and tomap this newly discovered land of unlimited mathematical strength.


References I

First-order LogicI Any good Logic text book.I Jon Barwise. An introduction to first-order logic. In J. Barwise, editor,

Handbook of Mathematical Logic, pp. 5–46. North-Holland, 1977.

Godel’s Incompleteness TheoremsI C. Smorynski. The incompleteness theorems. In J. Barwise, editor,

Handbook of Mathematical Logic, pp. 821–865. North-Holland, 1977.

Logical CalculiI A. S. Troelstra, H. Schwichtenberg. Basic Proof Theory, 2nd edition,

Cambridge University Press, 2000.


References II

Gentzen’s TheoremI M. E. Szabo, editor, Collected Papers of Gerhard Gentzen,

North-Holland, 1969.I R. Kahle. Gentzen’s Consistency Proof in Context. In R. Kahle and M.

Rathjen, editors, Gentzen’s Centenary, pp. 3–25. Springer, 2015.I M. Yasugi and K. Sangyo, editors, Memoirs of a Proof Theorist

[= G. Takeuti], World Scientific, 2003, Appendix B, pp. 121–134.

Proof Theory (textbooks)I K. Schutte, Proof Theory, Springer, 1977.I G. Takeuti, Proof Theory, 2nd edition, North-Holland, 1987 (reprint,

Dover, 2013).I J.-Y. Girard, Proof theory and logic complexity, Bibliopolis 1987.I W. Pohlers, Proof Theory, Springer, 2009.


References III

Proof Theory (“Applications”)I U. Kohlenbach, Applied Proof Theory, Springer, 2008.I S. Simpson, Subsystems of Second Order Arithmetic, 2nd ed., ASL and

Cambridge University Press, 2009.I H. Schwichtenberg and S. Wainer, Proofs and Computations, ASL and

Cambridge University Press, 2011.


Solomon Feferman, 1928–2016

Solomon Feferman, December 13, 1928–July 26, 2016

Sol Feferman was a distinguished proof theorist from Stanford Universitywho substantially contributed to modern proof theory, in particular, to theproof-theoretic analysis of (∆1

2-CA) + BI. This analysis was a landmark inthe extension of Gentzen-style ordinal analysis to impredicative theories.


cantor's naive set theory - uni-goettingen.de

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