1. The Unique Infinity of the Denumerable Reals Mathematics on the Edge of Quantum Reality

2. Dr. Brian L. Crissey Professor of Mathematics Professor of Mathematics North Greenville University, SC North Greenville University, SC Math/CS 1975 Math/CS 1975 Johns Hopkins Johns Hopkins

3. My Path Started with Math Started with Math Then Physics Then Physics Saw better opportunities in Computer Science Saw better opportunities in Computer Science But CS changed too quickly But CS changed too quickly Math seemed stable Math seemed stable Or so I thought Or so I thought

4. Simplification One of Mathematics’ Great Traditions 12 / 4= 3= 0

5. Today’s Intent To Simplify Transfinite Mathematics Down to… { φ } … the empty set א0א1א2א3…א0א1א2א3…

6. RATIONALS Chart of Numbers INTEGERS Finite PrecisionPotentially Infinite Precision 21 21/6 irrationals REALS

7. Infinite Periodic Precision Periodic Reals have infinitely long decimal expansions Example (1/7)10 ––0––0.142857142857142857142857… Where do they fit?

8. RATIONALS Repeating Expansions INTEGERS Finite PrecisionPotentially Infinite Precision 21 21/6 1/7 irrationals REALS

9. Eliminating Infinite Periodic Precision Change the base to the denominator Change the base to the denominator –(1/7) 10 = (0.1) 7 Radix is a presentation issue, not a characteristic of the number itself. Radix is a presentation issue, not a characteristic of the number itself.

10. RATIONALS Revised Chart of Numbers INTEGERS Finite PrecisionPotentially Infinite Precision 21 21/6 (0.1) 7 irrationals REALS

11. Are Irrationals Even Real? Leopold Kronecker 1823 - 1891 Georg Cantor’s Mentor Georg Cantor’s Mentor Strongly disputed Cantor’s inclusion of irrationals as real numbers Strongly disputed Cantor’s inclusion of irrationals as real numbers “My dear Lord God made all the integers. Everything else is the work of Man.” “My dear Lord God made all the integers. Everything else is the work of Man.”

12. Irrationals Never Reach The Real Number Line

13. What is a Real Number? Solomon Feferman 1928 – present Mathematician and philosopher at Stanford University Mathematician and philosopher at Stanford University Author of Author of –In the Light of Logic Reals are those numbers intended for measuring.

14. Influential Disciplines in the 20 th Century Physics Computer Science Quantum Theory Computability Has Math Integrated the New Knowledge?

15. Mathematical Minds from the Last Century Physics Physics  Quantum Theory  And the Limits of Measurability Computer Science Computer Science  Computability  And Enumeration  Time to Upgrade? Alan Turing Max Planck

16. From Quantum Physics Everything is energy Everything is energy Matter is perception of concentrated energy Matter is perception of concentrated energy “Particles” “Waves” Particle detector limit Smallest “particle”

17. Quantum Geometry A Quantum point occupies a non-zero volume A Quantum point occupies a non-zero volume Many implications Many implications “Particles” “Waves” A quantum “point”

18. Natural Units Max Planck suggested the establishment of “units of length, mass, time, and temperature that would … necessarily retain their significance for all times and all cultures, even extraterrestrial and extrahuman ones, and which may therefore be designated as natural units of measure.” Δ

19. Planck Precision Limits Quantum-scale granulation of reality Quantum-scale granulation of reality –Mass –Length –Time –Area –Volume –Density –Any measure

20. Planck Infinitesimals  L =  L = l pl = (hG/c 3 ) 1/2 = 10 -33 cm  m =  m = m pl = (hc/G) 1/2 = 10 -5 g  t =  t = t pl = (hG/c 5 ) 1/2 = 10 -43 s

21. Abraham Robinson, Mathematician 1918 – 1974 1918 – 1974 developed nonstandard analysis developed nonstandard analysis a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics.

22. Smallest Measurable Length  South Carolina South Carolina As a Proton is to a Planck length   is to a Proton

23.  The Quantum Limit   is the limit of measurability.  It is the quantum limit of  X in the differential quotient of Calculus.

24. Limited Real Precision  If real numbers are for measuring,  And measuring precision is limited by quantum mechanics,  Then measurable real numbers have limited precision.

25. A Lower Limit to Measurable Precision  L  L = 10 -35 m The “infinitesimal”

26. The Measurable Universe is Granular  V

27. Implication 1 Two real measures that differ by less than  are indistinguishable in our reality. If |r1 – r2| <  then r1 = r2

28. An Old Paradox Revisited 1.999… = 1 + 9 *.111… 1.999… = 1 + 9 *.111… 1.999… = 1+ 9 * 1/9 1.999… = 1+ 9 * 1/9 1.999… = 1 + 1 1.999… = 1 + 1 So 1.999… = 2 So 1.999… = 2 But at the quantum edge, But at the quantum edge, 2 – 1.999… = Δ ≠ 0 2 – 1.999… = Δ ≠ 0 So 2 ≠ 1.999… So 2 ≠ 1.999… 1.9999999999999999999999999999999999999999999999999999999

29. Classical 2:1 Point Paradox There are exactly as many points in a line segment of length 2 as there are in a line segment of length 1. There are exactly as many points in a line segment of length 2 as there are in a line segment of length 1. 2 1

30. Reality Math 2:1 Paradox Revisited The ratio of Δ- infinitesimals in a line segment of length 4 to those in a line segment of length 2 is 2:1. The ratio of Δ- infinitesimals in a line segment of length 4 to those in a line segment of length 2 is 2:1.

31. Classical Point-Density Paradox There are exactly as many points in a line segment of length 1 as there are on the entire real number line. There are exactly as many points in a line segment of length 1 as there are on the entire real number line.

32. Reality-Math Point- Density Resolved  Rounding b to the nearest Δ -integer shows that a:b is many-to-one, not 1- to-1

33. Pythagorus Good Old Pythagorus Good Old Pythagorus c 2 = a 2 + b 2 c 2 = a 2 + b 2 True for all right triangles True for all right triangles then and now and forever then and now and forever Maybe Maybe

34. Pythagorean Failures The hypotenuse of a quantum-scale isosceles right triangle, being a Δ – integer, cannot be irrational. The hypotenuse of a quantum-scale isosceles right triangle, being a Δ – integer, cannot be irrational. Three cases pertain. Three cases pertain.

35. Quantum Pythagorus Case 1 The hypotenuse is a truncated Δ – integer in a discontinuous triangle. The hypotenuse is a truncated Δ – integer in a discontinuous triangle. 9-9- 12.729… 9-9- 12.729… 9-9-12 9-9-12

36. Quantum Pythagorus Case 2 The hypotenuse is a rounded-up Δ – integer in a continuous triangle with overlap. The hypotenuse is a rounded-up Δ – integer in a continuous triangle with overlap. 9-9- 12.729… 9-9- 12.729… 9-9-13 9-9-13

37. Quantum Pythagorus Case 3 The triangle is continuous, The triangle is continuous, But the longest side is no hypotenuse because the triangle is not exactly right- angled. But the longest side is no hypotenuse because the triangle is not exactly right- angled.

38. Quantum Pythagorean Triples 3-4-5 3-4-5 5-12-13 5-12-13 Is there a minimal angle? Is there a minimal angle? 7-24-25? 7-24-25?

39. Quantum Geometry is Different A = ½ BH A = ½ BH H = 2A / B H = 2A / B A = 15 balls A = 15 balls B = 5 balls B = 5 balls But H ≠ 6 balls But H ≠ 6 balls

40. Geometry at the Quantum Edge of Reality Circles, when pressed against each other Circles, when pressed against each other Become hexagons Become hexagons

41. There are Three Regular Tesselations of the Plane Nature chooses the hexagon Nature chooses the hexagon

42. Natural Angles and Forms 60 º 60 º Equilateral triangles Equilateral triangles No right triangles at the quantum edge No right triangles at the quantum edge

43. Quantum Angles Straight lines intersect at fixed angles of 60 º and 120 º Straight lines intersect at fixed angles of 60 º and 120 º

44. Quantum Hexagonal Grid Cartesian coordinates can translate into quantum hexagon sites Cartesian coordinates can translate into quantum hexagon sites

45. What is a Quantum Circle? A quantum circle is a hexagon A quantum circle is a hexagon

46. Quantum Circles Not all circumferences exist Not all circumferences exist Not all diameters exist Not all diameters exist Not all “points” are equidistant from the center Not all “points” are equidistant from the center CircumferenceDiameterPi? 111.0 632.0 1252.4

47. Quantum Continuity Face-sharing may define continuity at the quantum edge Face-sharing may define continuity at the quantum edge But it fails as a function. But it fails as a function.

48. Quantum Discontinuity Greater slopes cause discontinuity at the quantum edge Greater slopes cause discontinuity at the quantum edge Only linear functions are continuous at the quantum edge Only linear functions are continuous at the quantum edge

49. Integration is Discrete Quantum Integration is discrete Quantum Integration is discrete The integral is a Δ- sum The integral is a Δ- sum Discontinuous functions are integrable. Discontinuous functions are integrable.

50. Quantum 3-D Structures What models will be useful in examining geometry at the quantum edge? What models will be useful in examining geometry at the quantum edge?

51. 3-D Quantum Geometry How do 3-D quanta arrange themselves naturally? How do 3-D quanta arrange themselves naturally?

52. Quantum Tesselation Spheres press together into 3-D tesselations. Spheres press together into 3-D tesselations.

53. A Real Partition Measurable reals have finite precision and are denumerable Measurable Speculative The Real Numbers Speculative reals may have infinite precision but are not computable

54. Measurable vs. Speculative The computation of √ 2 as a measure is truncated by Planck limits R = R m U R S √ 2 has infinite precision but never terminates.. 1.4142135623730950488016887242097… RsRs RmRm √ 2 * √ 2 returns no value, as the process never terminates.

55. Redefining Functions A real function must return a result A real function must return a result This is not a function : This is not a function : –Y(X) = { 1, if x is rational -1, if x is irrational } –Y(  ε R S ) will not terminate A function defined on Δ -integers, will always return a Δ- integer. A function defined on Δ -integers, will always return a Δ- integer.

56. Implication 2 Every real measure is an integral multiple of  and is thus is an integer. r ε Rm i ε Z such that r = i * Δ And i = └ r/Δ ┘ A E

57. Implication 3 Integers are denumerable, so measurable reals are denumerable. If cardinality (Z) = א0, then cardinality (Rm) = א0

58. Simplification Cardinality (Z) = C ardinality (R m ) = ∞

59. But What About the Speculative Reals Surely they are not denumerable R = R m U R S 1.4142135623730950488016887242097… RsRs RmRm

60. Irrationals Like √2 ε R s –1–1.41421356237309504880168872… Never deliver a usable result Or –T–T–T–They truncate to a rational approximation ε R m

61. Surely Pi is Irrational?  Pi: ratio of a circle’s circumference to its diameter  Circumference: measure of a circle’s perimeter  Diameter: The measure of a circle’s width

62.  Pi: is a ratio of a two measurable reals  Measurable reals are Δ - integers  So pi is rational

63. The Best Estimate of Pi Would be the measure of the greatest knowable circle Divided by the measure of its diameter

64. Estimating Rational Pi

65. What About Cantor? Is his work valid? Is his work valid? If not, what are the implications? If not, what are the implications?

66. Georg Cantor: A Sketch b. 1845 in St. Petersburg b. 1845 in St. Petersburg 1856 Moved to Germany 1856 Moved to Germany 1867 Ph.D. in Number Theory, University of Berlin 1867 Ph.D. in Number Theory, University of Berlin Professor, University of Halle Professor, University of Halle In and out of mental hospitals all his life In and out of mental hospitals all his life 1918 died in a sanatorium 1918 died in a sanatorium

67. Cantor’s Controversies Some Infinities are larger Some Infinities are larger Maybe Maybe Infinities can be completed Infinities can be completed Maybe Maybe Cardinalities can be operated upon Cardinalities can be operated upon Maybe Maybe

68. Discomfort with Actual Infinities Aristotle 384 BC -322 BC Greek Philosopher Greek Philosopher "The concept of actual infinity is internally contradictory" "The concept of actual infinity is internally contradictory" “Infinitum actu non datur” -Aristotle

69. Discomfort with Actual Infinities Henri Poincaré 1854-1912 Philosopher and Mathematician Philosopher and Mathematician Said that Cantor's work was a disease from which mathematics would eventually recover Said that Cantor's work was a disease from which mathematics would eventually recover “There is no actual infinity- Cantorians forgot that and fell into contradiction...”

70. Discomfort with Actual Infinities Ludwig Wittgenstein 1889-1951  Austrian philosopher  Rejected Cantor saying his argument “has no deductive content at all” Cantor’s ideas of uncountable sets and different levels of infinity are “a cancerous growth on the body of mathematics”

71. Discomfort with Cantor Alexander Alexandrovich Zenkin 1937-2006 1937-2006  “The third crisis in the foundations of mathematics was Georg Cantor’s cheeky attempt to actualize the Infinite.”

72. Discomfort with Cantor L.E.J. Brouwer 1881-1966 Dutch mathematician and philosopher Dutch mathematician and philosopher Founder of modern topology Founder of modern topology Attempted to reconstruct Cantorian set theory Attempted to reconstruct Cantorian set theory Cantor’s theory was “a pathological incident in the history of mathematics from which future generations will be horrified.”

73. Cantor’s Diagonal Enumerate the reals Enumerate the reals Output a non-denumerable real Output a non-denumerable real Conclusion: Conclusion: –Reals are not denumerable –So Cardinality(R) > Cardinality(Z) But Cantor produced a nonterminal output string, not a nondenumerable real But Cantor produced a nonterminal output string, not a nondenumerable real

74. Re-examining Cantor’s Diagonal Proof Cross-products of denumerable sets are denumerable Cross-products of denumerable sets are denumerable

75. Denumerable sets Integers Integers  Reals  Reals Input Strings Input Strings Characters Characters Words Words Sentences Sentences Paragraphs Paragraphs Procedures Procedures 1 2 3 4… 10 11 12… 99… 999… a b c… aa ab ac… zz… zzz… alpha beta… omega… All men are created equal… When in the course of human events…

76. Input-Driven Procedures Procedures are denumerable Procedures are denumerable Input strings are denumerable Input strings are denumerable are denumerable

77. Denumerating Cantor FUNCTION Cantor(nArray array of numbers) RETURN Number i, n Number; bArray(n) Array of Boolean; BEGIN // n is the length of the array rv = 1/2+ // set the initial return value to 1/2 n = nArray.length; // Initialize the values of boolean array to false. For i=1 to n str(i) = False; End Loop; // Process the in coming array. For i = 1 to n If nArray(n) is an integer bArray(i) = True; Else // Do nothing End If; If nArray(n) = rv Then // Find the next lowest value not in list Loop rv ++; Exit When bArray(rv) End Loop; If rv = n then // this will never happen print "Wow. The set of halves is the same size as the set of integers!!!" End If; End If; End Loop; RETURN rv; END; FUNCTION Cantor(nArray array of numbers) RETURN Number i, n Number; bArray(n) Array of Boolean; BEGIN // n is the length of the array rv = 1/2+ // set the initial return value to 1/2 n = nArray.length; // Initialize the values of boolean array to false. For i=1 to n str(i) = False; End Loop; // Process the in coming array. For i = 1 to n If nArray(n) is an integer bArray(i) = True; Else // Do nothing End If; If nArray(n) = rv Then // Find the next lowest value not in list Loop rv ++; Exit When bArray(rv) End Loop; If rv = n then // this will never happen print "Wow. The set of halves is the same size as the set of integers!!!" End If; End If; End Loop; RETURN rv; END; Somewhere in the list of all possible procedures is Cantor’s procedure to generate a non- denumerable real

78. Cantor’s Failed Diagonal Argument Cantor’s non-enumerated real Cantor’s non-enumerated real Is just a process output Is just a process output Matched digit by digit by the output of the correct enumerated procedure Matched digit by digit by the output of the correct enumerated procedure There is no non- enumerated real There is no non- enumerated real CANTOR 2.32514…

79. Implication 5 Measurable reals are denumerable, and speculative reals are denumerable, so all reals are denumerable. Cardinality (Z) = א0 = cardinality (Rm) = cardinality (Rs) = ∞

80. If Cantor’s Wrong… “Cantor’s [diagonal] theorem is the only basis and acupuncture point of modern meta-mathematics and axiomatic set theory in the sense that if Cantor’s famous diagonal proof of this theorem is wrong, then all the transfinite … sciences fall to pieces as a house of cards.” Alexander Zenkin

81. Implications  According to truth tables  False implies anything is true  So if Cantor was wrong, we have falsely implied some conclusions

82. The Continuum Hypothesis Hilbert 1900 Hilbert 1900 First of 23 great Unanswered Math Questions First of 23 great Unanswered Math Questions “Does there exist a cardinal between א 0 & c?” “Does there exist a cardinal between א 0 & c?” λ between א 0 and c א 0 ≤ λ ≤ c ?

83. Implication 6 The Continuum Hypothesis can be confirmed. א0 = c = ∞ There is no cardinal between א 0 and c because they are equal.

84. David Hilbert “No one shall drive us from the paradise Cantor created for us.” “No one shall drive us from the paradise Cantor created for us.”

85. Driven from Paradise? Is the Cantorian Church of PolyInfinitism in need of reform?

86. The ¯ Theses  There is but one infinity  Reals are denumerable  א 0 = א 1 = א 2 = א 3 … = ∞  Cardinality(R) = c = ∞ = C(Z)  There are no right triangles at the Quantum Edge  Geometry changes at the Quantum Edge  What else has kicked the bucket?.99

87. The “Kicked the Bucket” List  There are infinities of infinities  Reals are not denumerable  א 0 < א 1 < א 2 < א 3 …  Cardinality(R) = c = 2 א 0 > א 0 = C(Z)  Universality of Pythagorean Theorem  Metamathematics  Transfinite Mathematics  Axiomatic Set Theory…

88. Conclusion We have graduated into We have graduated into –The Quantum Mathematical Universe Many things may change Many things may change

89. The Great Circle Math and Physics Math and Physics Computer Science Computer Science CS changed too quickly CS changed too quickly Math seemed stable Math seemed stable Now I’m not so sure. Now I’m not so sure. Perhaps I’ll head back to CS Perhaps I’ll head back to CS –Where things don’t change so much… א 5 א 0 א 5

90. A New Beginning