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Exploring the Isomorphism of pq and tq Systems with Integer Operations

This study investigates the isomorphism and equivalence of pq and tq systems to the addition and multiplication of positive integers, respectively. By defining specific axioms and rules, the research explores the relationship between integer operations and logical frameworks in mathematical systems. The methodology includes rules for establishing mathematical identities and a focus on composite and prime numbers, presenting a unique angle on classical arithmetic concepts. The findings aim to clarify the structural similarities between these systems.

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Exploring the Isomorphism of pq and tq Systems with Integer Operations

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

  2. Is the tq-system isomorphic to multiplication of positive integers? AXIOMS: xt - q x (Defines x ·1) RULE: IF xt y q z THEN xty -q zx (IF x · y = z THEN x ·(y+1) = z+x) pq-system IF xp y q z THEN xp 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

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

  4. New rule: Capturing compositeness

  5. New rule: Capturing compositeness

  6. New rule: Capturing compositeness

  7. New rule: Capturing compositeness

  8. New rule: Capturing compositeness

  9. 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 = - - ? - - - - - t - - - q - - - - - - - - - t - - - - q - - - - - - 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?

  10. New rule: Capturing primes

  11. New rule: Capturing primes

  12. New rule: Capturing primes How did we get these primes?

  13. New rule: Capturing primes Rule: IF x - t y - q z THEN C z Proposed Rule: IF C x is not a theoremTHEN P x Permitted typographical operations 1. Reading, recognizing symbols2. Writing down symbols3. Copying symbols4. Erasing symbols5. Comparing symbols, determining identity6. Remembering established theorems

  14. (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)

  15. - - - - - tq (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 - - - - - tq - - - - - - t - - q - - - - x y z - - M Rule Machine Write down contents of box M in box z Copy contents of box x to box M Write down - in box y

  16. New rule: Capturing primes Rule: IF x - t y - q z THEN C z Proposed Rule: IF C z is not a theoremTHEN 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 zrule use only typographical operations? Does the P zrule use only typographical operations?

  17. Figure and Ground Sky and Water, MC Escher

  18. Figure and Ground (Fughetta by JS Bach)

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