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Welcome to. Gödel. Escher. Bach. Tuesday edition, February 27, 2001. News. Seminar Wed Feb 28, 4:00 PM Jepson 109 The Poincaré Conjecture Mike Kerckhove discusses whether we can determine whether our universe is in the shape of a sphere or a doughnut.

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Gödel

Escher

Bach

Tuesday edition, February 27, 2001

News

Seminar Wed Feb 28, 4:00 PM Jepson 109 The Poincaré ConjectureMike Kerckhove discusses whether we can determine whether our universe is in the shape of a sphere or a doughnut

Exam II Almost everyone has taken it. Unlikely to be returned before next Tuesday.

(Symbols of previous systems)

MIU-System: M I U

pq-System: - p q

Atom particle: -

Atoms: - or - - or - - - or ...

(Symbols of previous systems)

MIU-System: M I U

pq-System: - p q

Operation: p

atoms

- - - p - -

Example:

p

Symbols of the Propositional Calculus

(Symbols of previous systems)

MIU-System: M I U

pq-System: - p q

Operation: q

atom

- - - p - -

q - -

Example:

(Symbols of previous systems)

MIU-System: M I U

pq-System: - p q

tq-System: - t q

C-System: - t q C

Prefix operation: C

C - - -

Example:

How to classify them?

< >

P Q R ’

^ v ~

[ ]

Atom particles

Example:

P

All atom particles are atoms

Symbols of the Propositional Calculus

How to classify them?

< >

P Q R ’

^ v ~

[ ]

Atom particles

Atom suffix

Example:

P

All atoms are well-formed

How to classify them?

< >

P Q R ’

^ v ~

[ ]

Atom particles

Suffix

Prefix op

Formation Rule #1: If x is well-formed, then so is ~x

Example:

~

x

P

How to classify them?

< >

P Q R ’

^ v ~

[ ]

Atom particles

Suffix

Infix ops

Prefix op

Formation Rules #2: If x and y are well-formed, then so is <x^y>

Example:

<x^y>

<P^~Q>

How to classify them?

< >

P Q R ’

^ v ~

[ ]

Infix op start

Infix op stop

Atom particles

Suffix

Infix ops

Prefix op

Procedure: Push

Procedure: Pop

Fantasy Rule

Example:

<P^~Q>

(Study Question 3)What’s so recursive about the Formation Rules?

Formation Rule #1: If x is well-formed, then so is ~x

Formation Rule #1: If is well-formed, then so is

~

~

(Study Question 3)What’s so recursive about the Formation Rules?

Formation Rule #1: If x is well-formed, then so is ~x

Formation Rule #1: If is well-formed, then so is ~

Formation Rule #1: If is well-formed, then so is

~

(Study Question 3)What’s so recursive about the Formation Rules?

Formation Rule #1: If x is well-formed, then so is ~x

Formation Rule #1

P

~P

~~P

Study Question 7Which of the strings are well-formed?

Formation Rule #1: If x is well-formed, then so is ~xFormation Rules #2: If x and y are well-formed, then so is <x^y>Formation Rules #3: If x and y are well-formed, then so is <xvy>Formation Rules #4: If x and y are well-formed, then so is <x y>

Formation Rule #0: All atoms are well-formed atoms

<~P>

< x >

NOT well-formed

<P^Q>

<x^y>

WELL-formed

<x^y^z>

<P^Q^R>

NOT well-formed

Study Question 7Which of the strings are well-formed?

Formation Rule #1: If x is well-formed, then so is ~xFormation Rules #2: If x and y are well-formed, then so is <x^y>Formation Rules #3: If x and y are well-formed, then so is <xvy>Formation Rules #4: If x and y are well-formed, then so is <xy>

Formation Rule #0: All atoms are well-formed atoms

<<P v<QR>>^<~Pv~R’>>

<<xv<xy>>^<~xv~y>>

<<xv y>^< xv y >>

< x^ y >

WELL-formed

1. I think; therefore I am.- Rene Descartes

2. Good judgment comes from experience, and often experience comes from bad judgment.” [Hint: ignore “often”]- Rita Mae Brown

3. Like it or not, here it comes.- Anonymous

4. If you are a retiree, use your age on the annuity starting date,...

but if your annuity starting date was after 1997 and the payments are for your life and that of your beneficiary, use your combined ages on the annuity starting date.

- One of the more comprehensible sections of the US tax code