Godel’s Proof¨websupport1.citytech.cuny.edu/Faculty/tjohnstone/talks/godel.pdf · natural...

Godel’s Proof

Victoria Gitman and Thomas Johnstone

New York City College of Technology, CUNY

[email protected]://websupport1.citytech.cuny.edu/faculty/vgitman

[email protected]

March 17, 2009

Victoria Gitman and Thomas Johnstone (CUNY) Godel’s Proof March 17, 2009 1 / 21

Mathematical Logic

Mathematical Logic

Logic: Study of reasoning

Mathematical logic:

study of type of reasoning done by mathematicians

examines the methods used by mathematicians

Mathematics, as opposed to other sciences, uses proofs instead of observations.

impossible to prove all mathematical laws

certain first laws, axioms, are accepted without proof

the remaining laws, theorems, are to be proved from axioms

How do we accept certain axioms?

How do we choose reasonable axioms? Non-contradictory axioms? Powerfulaxioms?

What constitutes a proof from a given set of axioms? Which rules do we have dofollow at each step?


Mathematical Logic

Gottlob Frege (1848-1925)

In his Begriffsschrift (1879), Frege

introduces symbolism for predicate logic

invents quantified variables: “for all”,and “there exists”

makes iterations of ∀ and ∃ understandable

“Every boy loves some girl” vs. “Some girl is loved by all boys”

invents axiomatic predicate logic

In his Grundgesetze der Arithmetik(1893, 1903), Frege attempts to axiomatize thetheory of sets.


Mathematical Logic

Frege’s Set Building Axiom

“For any formal criterion, there exists a set whose members are those objects (andonly those objects) that satisfy the criterion.”

Frege’s axioms allows us to build various sets:

the set N of all natural numbers

the set R = {x : x is a real number}the set I = {x : x is an infinite set}the set of all sets, V = {x : x = x}

Some sets are members of themselves, while others are not!

Consider the set B of all objects that are not members of themselves, i.e.

B = {x : x /∈ x}

Question: What’s the problem with B?


Mathematical Logic

Russell’s Paradox (1901)

Bertrand Russell (1872-1970) discovers that Frege’s axioms lead to a contradiction.

The key ideas:

The set B = {x : x /∈ x} cannot exist.

Self-reference: “x is not an element of itself”

Similar to the Liar Paradox (Epimenides, 400 BC):

This sentence is false!

Russell fixed Frege’s system using type theory.

This led to the ComprehensionAxiom and Zermelo-Fraenkel set theory.


Formal Mathematics

Formal Mathematics

The 19th century work of Frege, Russell, Hilbert, Peano, Cantor, etc. leads todevelopment of formal systems:

A formal system consists of

A formal language

Axioms

Rulesof inference (how to conclude theorems from axioms)


Formal Mathematics

Hilbert’s Program (1921)

David Hilbert (1862-1943) aimed to provide a secure foundation for mathematics.

Two Key Questions

Consistency: How do we know that contradictory consequences cannot be proved fromthe axioms?

Completeness: What if there are statements that cannot be decided by the axioms?

“No one shall expel us from the paradise that Cantor has created for us.”....Hilbert


Formal Mathematics

Hilbert’s Program (continued...)

Translate all mathematics into a formallanguage and demonstrate “by finitary means” that

Peano Axioms (PA) for Number Theory,

Zermelo-Fraenkel (ZF) Axioms for Set Theory,

Euclidian Axioms for Geometry,

Principia Mathematica (PM) Axioms,

are consistent and complete!

What are Hilbert’s “ finitary means”?

Problem:

natural numbers, the universe of sets, the Euclidianspace are not finite!

Solution: Strip away the meaning of mathematical assertions!


Formal Mathematics

Hilbert’s Program (continued...)

A mathematical assertion can be viewed in two fundamentally different ways:

as a sentence, namely a sequence of letters and symbols, or

as the meaning of the sentence.

Advantages of the syntactical view:

Mathematical concepts are very abstract

A sentence, studied as a syntactical object with no meaning, is very concrete.

Proofs are finite sequences of sentences that follow a few simple rules.

Provability can be studied without any understanding of the underlying subject.

“Mathematics is a game played according to certain simple rules with meaninglessmarks on paper.” ...Hilbert


Formal Mathematics

Crash Course in Formal Mathematics

1) Logical symbols:

Equality: =

Boolean connectives: ∨, ∧, ¬,→Quantifiers: ∃, ∀

2) Functions, relations, and constants symbols: specific to subject

Number Theory: +, ·, <, 0, 1

Set Theory: ∈Group Theory: ◦, −1, e

3) Variables:x1, x2, x3, x4, . . . infinitely many!

4) Punctuation symbols:(, )

For notational convenience, we will write x , y , z instead of x1, x2, x3, respectively.


Formal Mathematics

Writing Formal Mathematics: Formulas

Examples of formulas in Number Theory

x is even: even(x) := ∃y y + y = x

x divides y : x |y := ∃z z · x = y

x is prime: prime(x) := (∀y (y |x → (y = 1 ∨ y = x)) ∧ ¬x = 1)

3x = y : suggestions? (problem is that definition is recursive)

xy = z: (same problem)

3 is even: ∃x (x = (1 + 1) + 1 ∧ even(x))

There are infinitely many primes: ∀x∃y (y > x ∧ prime(y))

Every even number > 2 is the sum of two primes:∀x ((x > 1 + 1 ∧ even(x))→ ∃y∃z ((prime(y) ∧ prime(z)) ∧ x = y + x)))


Formal Mathematics

Formulas (continued...)

So what is a formula?

Slogan: A formula, should if translated to English, correspond to a complete sentence.

Really, after stripping away any meaning:

A formula is a string of symbols built according to a finite set of simple rules.

This string is a formula: (why?)

∃z(z > 0 ∧ x + y = z)

This string is not a formula: (why not?)

∀x(y ∧ ∀z z > 0)

What are the rules?


Formal Mathematics

Formula Witnessing Sequences

Recursive rules for building formulas:

Equality statements are formulas: x = y , x + y = z · zLess than statements are formulas: x + 1 < z

Boolean combinations of formulas are formulas:if ϕ and ψ are formulas, then so are(ϕ ∧ ψ), (ϕ ∨ ψ), ¬ϕ, (ϕ→ ψ).

A formula with a quantifier-variable pair attached in front is a formula:if i is any natural number and ϕ is a formula and , then so are ∃xi ϕ, ∀xi ϕ.

Nothing else is a formula

This recursive definition gives rise to formula witnessing sequences.


Formal Mathematics

The Peano AxiomsAxiomatization of Number Theory proposed by Giuseppe Peano (1858-1932).

Peano AxiomsAddition and Multiplication

∀x∀y∀z (x + y) + z = x + (y + z) (associativity of addition)

∀x∀y x + y = y + x (commutativity of addition)

∀x∀y∀z (x · y) · z = x · (y · z) (associativity of multiplication)

∀x∀y x · y = y · x (commutativity of multiplication)

∀x∀y∀z x · (y + z) = x · y + x · z. (distributive law)

∀x (x + 0 = x ∧ x · 1 = x) (additive and multiplicative identity)


Formal Mathematics

Peano Axioms (continued)Order

∀x∀y∀z ((x < y ∧ y < z)→ x < z) (the order is transitive)

∀x ¬x < x (the order is anti-reflexive)

∀x∀y ((x < y ∨ x = y) ∨ y < x) (any two elements are comparable)

∀x∀y∀z (x < y → x + z < y + z) (order respects addition)

∀x∀y∀z ((0 < z ∧ x < y)→ x · z < x · z) (order respects multiplication)

∀x∀y (x < y ↔ ∃z (z > 0 ∧ x + z = y))

∀x (x ≥ 0 ∧ (x > 0→ x ≥ 1)) (the order is discrete)

Induction SchemeFor every formula ϕ(x) we have

(ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1)))→ ∀xϕ(x)

Hilbert’s question: Are the Peano Axioms consistent? Are they complete?


Formal Mathematics

The Group Theory Axioms: An Easy Example

Language: ◦, −1, e

Group Theory Axioms∀x∀y∀z x ◦ (y ◦ z) = (x ◦ y) ◦ z (associativity)

∀x (e ◦ x = x ∧ x ◦ e = x) (e is the identity)

∀x x ◦ x−1 = e (−1 is the inverse)

QuestionAre the group theory axioms consistent?


Formal Mathematics

Group Theory Axioms: (continued...)

Hilbert would like:

Z4◦ e a b ce e a b ca a b c eb b c e ac c e a b

QuestionAre the group theory axioms complete?


Formal Mathematics

Propositional LogicLanguage:

Variables: A, B, C, . . .Boolean connectives: ∧, ∨, ¬,→Punctuation symbols: (, )

Think of these variables as standing in for any sentence:

A =“Bush is a great public speaker”B =“It is going to rain tomorrow”C =“This talk is boring”

Rules for building formulas:A variable is a formula.Boolean combinations of formulas are formulas: (A→ B), ¬(A ∨ B)

The Rule of Inference is Modus Ponens: From A and A→ B, infer B.

Propositional Logic Axioms(A ∨ A)→ A

A→ (A→ B)

(A ∨ B)→ (B ∨ A)

(A→ B)→ ((C ∨ A)→ (C ∨ B))


Formal Mathematics

Propositional Logic is Consistent

The following formulas are provable from the axioms:

A→ A

(A ∨ ¬A)

(A ∧ ¬A)→ B

The axioms are inconsistent if for some formula A, both A and ¬A are provable.

It suffices to find a single formula that is not provable!

Hilbert’s Strategy:

Find a structural, syntactical property that every derivable formula has.

Show that not every possible formula has that property.

What is the desired property?

The formula is a tautology, i.e. “true in all possible worlds”

Theorem (Hilbert, 1905)Propositional logic is consistent.


Formal Mathematics

Presburger Arithmetic

Arithmetic without multiplication:

Presburger AxiomsAddition

∀x ¬0 = x + 1

∀x∀y (x + 1 = y + 1→ x = y)

∀x x + 0 = x

∀x∀y (x + y) + 1 = x + (y + 1)

Induction SchemeFor every formula ϕ(x) we have

(ϕ(0) ∧ ∀x (ϕ(x)→ ϕ(x + 1)))→ ∀xϕ(x)

Mojzesz Presburger (1904-1943) showed in 1929 using finitary arguments thatPresburger Arithmetic is consistent and complete!


Formal Mathematics

Godel ends Hilbert’s ProgramTheorem (Godel, 1931)The Peano Axioms are not complete. In fact, any “reasonable” collection of axioms forNumber Theory or Set Theory is necessarily incomplete.

Theorem (Godel, 1931)No proof of the consistency of the Peano Axioms can be given by “finitary means”.


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