Big Picture Much of the course so far

• David Evans

• http://www.cs.virginia.edu/evans

Big Picture

• Much of the course so far:

• Getting comfortable with recursive definitions
• Learning to write a program to do anything

• Today:

• Getting uncomfortable with recursive definitions

• Monday:

• Convincing you there are some things no one can write a program to do!

Mechanized Reasoning

Mechanical Reasoning

• Aristotle (~350BC): Organon

• We can explain logical deduction with rules of inference (syllogisms)
• Every B is A
• C is B
•  C is A
• Every human is mortal.
• Gödel is human.
• Gödel is mortal.

More Mechanical Reasoning

• Euclid (~300BC): Elements

• We can reduce geometry to a few axioms and derive the rest by following rules

• Newton (1687): Philosophiæ Naturalis Principia Mathematica

• We can reduce the motion of objects (including planets) to following axioms (laws) mechanically

Mechanical Reasoning

• Late 1800s – many mathematicians working on codifying “laws of reasoning”

• George Boole, Laws of Thought
• Augustus De Morgan
• Whitehead and Russell

All true statements about number theory

Perfect Axiomatic System

Incomplete Axiomatic System

Inconsistent Axiomatic System

Principia Mathematica

• Whitehead and Russell (1910– 1913)

• Three Volumes, 2000 pages

• Attempted to axiomatize mathematical reasoning

• Define mathematical entities (like numbers) using logic
• Derive mathematical “truths” by following mechanical rules of inference
• Claimed to be complete and consistent
• All true theorems could be derived
• No falsehoods could be derived

Russell’s Paradox

• Some sets are not members of themselves

• set of all Jeffersonians

• Some sets are members of themselves

• set of all things that are non-Jeffersonian

• S = the set of all sets that are not

• members of themselves

• Is S a member of itself?

Russell’s Paradox

• S = set of all sets that are not members of themselves

• Is S a member of itself?

• If S is an element of S, then S is a member of itself and should not be in S.
• If S is not an element of S, then S is not a member of itself, and should be in S.

Ban Self-Reference?

• Principia Mathematica attempted to resolve this paragraph by banning self-reference

• Every set has a type

• The lowest type of set can contain only “objects”, not “sets”
• The next type of set can contain objects and sets of objects, but not sets of sets

Russell’s Resolution?

• Set ::= Setn

• Set0 ::= { x | x is an Object }

• Setn ::= { x | x is an Object or a Setn - 1 }

• S: Setn

• Is S a member of itself?

Epimenides Paradox

• Epidenides (a Cretan):

• “All Cretans are liars.”

• Equivalently:

• “This statement is false.”

Gödel’s Solution

• All consistent axiomatic formulations of number theory include undecidable propositions.

• (GEB, p. 17)

• undecidable – cannot be proven either true or false inside the system.

Quiz

Kurt Gödel

• Born 1906 in Brno (now Czech Republic, then Austria-Hungary)

• 1931: publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions of Principia Mathematica and Related Systems)

Gödel’s Theorem

• In the Principia Mathematica system, there are statements that cannot be proven either true or false.

Gödel’s Theorem

• In any interesting rigid system, there are statements that cannot be proven either true or false.

Gödel’s Theorem

• All logical systems of any complexity are incomplete: there are statements that are true that cannot be proven within the system.

Proof – General Idea

• Theorem: In the Principia Mathematica system, there are statements that cannot be proven either true or false.

• Proof: Find such a statement

Gödel’s Statement

• G: This statement of number theory does not have any proof in the system of Principia Mathematica.

• G is unprovable, but true!

Gödel’s Proof

• G: This statement of number theory does not have any proof in the system of PM.

• If G were provable, then PM would be inconsistent.

• If G is unprovable, then PM would be incomplete.

• PM cannot be complete and consistent!

Finishing The Proof

• Turn G into a statement in the Principia Mathematica system

• Is PM powerful enough to express “This statement of number theory does not have any proof in the system of PM.”?

How to express “does not have any proof in the system of PM”

• What does it mean to have a proof of S in PM?

• There is a sequence of steps that follow the inference rules that starts with the initial axioms and ends with S

• What does it mean to not have any proof of S in PM?

• There is no sequence of steps that follow the inference rules that starts with the initial axioms and ends with S

Can PM express unprovability?

• There is no sequence of steps that follow the inference rules that starts with the initial axioms and ends with S

• Yes: (using Scheme-ified Gödel notation)

• (unprovable? x) = (not (provable? x))
• (provable? x) = (exists (y) (proves y x))
• (proves x y) =
• (and (valid-proof-steps x)
• (eq? (apply-proof x initial-axioms) y)))

Can we express “This statement of number theory”

• Yes!

• That’s the point of the TNT Chapter in GEB

• We can write turn every statement into a number, so we can turn “This statement of number theory does not have any proof in the system of PM” into a number

Gödel’s Proof

• G: This statement of number theory does not have any proof in the system of PM.

• If G were provable, then PM would be inconsistent.

• If G is unprovable, then PM would be incomplete.

• PM cannot be complete and consistent!

Generalization

• All logical systems of any complexity are incomplete:

• there are statements that are true that cannot be proven within the system.

Practical Implications

• Mathematicians will never be completely replaced by computers

• There are mathematical truths that cannot be determined mechanically
• We can build a computer that will prove only true theorems about number theory, but if it cannot prove something we do not know that that is not a true theorem.

Charge

• Next class:

• Does not being able to prove things mechanically have anything to do with not being able compute things?
• What is the equivalent to the Gödel sentence for computation?