Computation, Computers, and Programs Godel's incompleteness theorem

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CS20a Models (Nov 14, 2002)

• Alternative models
• Primitive recursive functions
• Partial recursive functions
• Recursion theorem
• Rices theorem
• Lambda Calculus
• Recursion theorem
• Rices theorem
• Arithmetic
• Recursion theorem
• Godels incompleteness theorem (aka Rices
  theorem)

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Arithmetic

• Lets do the same for arithmetic
• Arithmetic is
• Operators , -, , /,
• First-order logic
• First, lets define logic

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Defining logics

• A logic is defined in three parts
• Syntax
• Define what is true
• Define derivation procedures

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Defining the syntax
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Standard syntax definition
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Semantics
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Extending the truth assignment
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Some definitions
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A proof (Pierces Law)
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Derivations

• Truth tables are hard to use
• We want a mechanical method for proving
  propositions
• Use sequents and truth judgments

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Sequents
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Derivations

• There are two kinds of inference rules
• Introduction rules operate on the right of the
  turnstile
• Elimination rules operate on the left of the
  turnstile
• The base axiom

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Introduction rules, part I
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Introduction rules, part II
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Elimination rules
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Rule table
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Proving the law of excluded middle

• Every proposition is either true or false

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Pierces law
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Currying
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First-order logic
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Some examples
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Adding rules for FOL
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A quantifier DeMorgan law
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Another proof
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Formalizing arithmetic
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Indexing the arithmetic formulas
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Defining substitution
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Fixpoint theorem for arithmetic
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Fixpoint proof
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Godels Incompleteness Theorem