2. Is the pq-system equivalent to addition? Is the pq-system isomorphic to addition of positive integers? AXIOMS: x p - q x - RULE: IF x p y q z THEN x p y - q z - x p - q x - - p - q - - 1 + 1 = 2 x p - q x - - - p - q - - - 2 + 1 = 3 - p - - q - - - 1 + 2 = 3 - p - - - q - - - - 1 + 3 = 4 - - p - - q - - - - 2 + 2 = 4 x p - q x - - - - p - q - - - - 3 + 1 = 4 - - - p - - q - - - - - 3 + 2 = 5 - p - - - - q - - - - - 1 + 4 = 5 - - p - - - q - - - - - 2 + 3 = 5 x p - q x - - - - - p - q - - - - - 4 + 1 = 5 (IF x + y = z THEN x + y+1 = z+1) 1 2 3 4 12341234 (Defines x + 1)

3. Is the tq-system isomorphic to multiplication of positive integers? AXIOMS: RULE: IF x t y q z THEN x t y - q z x (IF x · y = z THEN x ·(y+1) = z+x) (Defines x ·1) x t - q x pq-system IF x p y q z THEN x p y - q z - IF x + y = z THEN x + y+1 = z+1 - p - q - - 1 + 1 = 2 - p - - q - - - 1 + 2 = 3 - p - - - q - - - - 1 + 3 = 4 - p - - - - q - - - - - 1 + 4 = 5

4. Is the tq-system isomorphic to multiplication of positive integers? AXIOMS: RULE: IF x t y q z THEN x t y - q z x (IF x · y = z THEN x ·(y+1) = z+x) (Defines x ·1) x t - q x pq-system IF x p y q z THEN x p y - q z - IF x + y = z THEN x + y+1 = z+1 - p - q - - 1 + 1 = 2 - p - - q - - - 1 + 2 = 3 - p - - - q - - - - 1 + 3 = 4 - p - - - - q - - - - - 1 + 4 = 5 tq-system IF x t y q z THEN x t y - q z x IF x · y = z THEN x · (y+1) = z+x - t - q - 1 · 1 = 1 - t - - q - - 1 · 2 = 2 - t - - - q - - - 1 · 3 = 3 - t - - - - q - - - - 1 · 4 = 4

5. New rule: Capturing compositeness

10. RULE: IF x - t y - q z THEN C z IF (x+1) · (y+1) = z THEN z is composite Test: - - - - - - (6) is a composite, so C - - - - - - should be true z =?- - - x =?y =?- - - - Why not: IF x t y q z THEN C z IF x · y = z THEN z is composite Test: x = - y = - z =?-C - is true? - - - t - - - - q - - - - - - - - - t - - - q - - - - - - -

11. New rule: Capturing primes

13. How did we get these primes?

14. New rule: Capturing primes Rule: IF x - t y - q z THEN C z Proposed Rule: IF C x is not a theorem THEN P x Permitted typographical operations 1. Reading, recognizing symbols 2. Writing down symbols 3. Copying symbols 4. Erasing symbols 5. Comparing symbols, determining identity 6. Remembering established theorems

15. (How do these operations work?) Permitted typographical operations 1. Reading, recognizing symbols 2. Writing down symbols 3. Copying symbols 4. Erasing symbols 5. Comparing symbols, determining identity 6. Remembering established theorems Is - - t - - - q - - - - - - a theorem? Method 1: 2 times 3 = 6, so it IS a theorem Method 2: - - - - - - is the same as (- - -) and (- - -) Method 3: - - p - q - - Axiom (x t - q x x = - -) - - p - - - q - - - - Rule (x t y q z x p y - q z x) - - p - - - q - - - - - - Rule (x t y q z x p y - q z x)

16. - - t - - q - - - - (Is the application of a rule permitted?) Permitted typographical operations 1. Reading, recognizing symbols 2. Writing down symbols 3. Copying symbols 4. Erasing symbols 5. Comparing symbols, determining identity 6. Remembering established theorems t q x y z - - - - -t q - - - - - Rule Machine M Copy contents of box x to box M - Write down - in box y - Write down contents of box M in box z -

17. New rule: Capturing primes Rule: IF x - t y - q z THEN C z Proposed Rule: IF C z is not a theorem THEN P z Permitted typographical operations 1. Reading, recognizing symbols 2. Writing down symbols 3. Copying symbols 4. Erasing symbols 5. Comparing symbols, determining identity 6. Remembering established theorems Does the C z rule use only typographical operations? Does the P z rule use only typographical operations?

18. Figure and Ground Sky and Water, MC Escher

19. Figure and Ground (Fughetta by JS Bach)