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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
1
2
3
4
(Defines x + 1)
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
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 xIF 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
New rule: Capturing compositeness
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing compositeness
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing compositeness
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing compositeness
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing compositeness
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing compositeness
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 - - - - - -
-
New rule: Capturing primes
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing primes
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
New rule: Capturing primes
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
How did we get these primes?
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 symbols2. Writing down symbols3. Copying symbols4. Erasing symbols5. Comparing symbols, determining identity6. Remembering established theorems
(How do these operations work?) Permitted typographical operations
1. Reading, recognizing symbols2. Writing down symbols3. Copying symbols4. Erasing symbols5. Comparing symbols, determining identity6. 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)
- - t - - q - - - -
(Is the application of a rule permitted?) Permitted typographical operations
1. Reading, recognizing symbols2. Writing down symbols3. Copying symbols4. Erasing symbols5. Comparing symbols, determining identity6. 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
- -
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 symbols2. Writing down symbols3. Copying symbols4. Erasing symbols5. Comparing symbols, determining identity6. Remembering established theorems
Does the C z rule use only typographical operations?Does the P z rule use only typographical operations?
Figure and Ground
Sky and Water, MC Escher
Figure and Ground
(Fughetta by JS Bach)
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![bw)># >] 8 PS ])...WfG 2Wf WfcWf q) %)Pq J 3fWvcP Wv P EW9P cf9 q) %)Pq J 3fWvc c fqP)f9P3 Wf v 9P3 P ) 9jq9P3 vj9P)jjU #v f Pq))% %) )Pq c 7) G 2Wf Wv vqP WvYf) WfG9P3 2Wf cf9 q)](https://img.pdfslide.us/doc/110x75/61345670dfd10f4dd73baaa2/bw-8-ps-wfg-2wf-wfcwf-q-pq-j-3fwvcp-wv-p-ew9p-cf9-q-pq.jpg)
