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## The Unique Infinity of the Denumerable Reals

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**The Unique Infinityof the Denumerable Reals**Mathematics on the Edge of Quantum Reality**Dr. Brian L. Crissey**• Professor of Mathematics • North Greenville University, SC • Math/CS 1975 • Johns Hopkins**My Path**• Started with Math • Then Physics • Saw better opportunities in Computer Science • But CS changed too quickly • Math seemed stable • Or so I thought**Simplification**One of Mathematics’ Great Traditions 12 / 4 = 3 = 0**Today’s Intent**א0א1א2א3 … To Simplify Transfinite Mathematics Down to… { φ } … the empty set**Chart of Numbers**Potentially Infinite Precision Finite Precision REALS RATIONALS INTEGERS 21/6 irrationals 21**Infinite Periodic Precision**• Periodic Reals have infinitely long decimal expansions • Example (1/7)10 • 0.142857142857142857142857… • Where do they fit?**Repeating Expansions**1/7 Potentially Infinite Precision Finite Precision REALS RATIONALS INTEGERS 21/6 irrationals 21**Eliminating Infinite Periodic Precision**• Change the base to the denominator • (1/7)10 = (0.1) 7 • Radix is a presentation issue, not a characteristic of the number itself.**Revised Chart of Numbers**(0.1)7 Potentially Infinite Precision Finite Precision REALS RATIONALS 21/6 INTEGERS irrationals 21**Are Irrationals Even Real?**Leopold Kronecker 1823 - 1891 • Georg Cantor’s Mentor • 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.”**What is a Real Number?**Reals are those numbers intended for measuring. Solomon Feferman 1928 – present • Mathematician and philosopher at Stanford University • Author of • In the Light of Logic**Influential Disciplinesin the 20th Century**Computer Science Physics QuantumTheory Computability Has Math Integratedthe New Knowledge?**Mathematical Mindsfrom the Last Century**• Physics • Quantum Theory • And the Limits of Measurability • Computer Science • Computability • And Enumeration • Time to Upgrade? Alan Turing Max Planck**From Quantum Physics**Everything is energy Matter is perception of concentrated energy Particle detector limit Smallest “particle” “Particles” “Waves” Δ**Quantum Geometry**A Quantum point occupies a non-zero volume Many implications A quantum “point” “Particles” “Waves” Δ**Natural Units**Max Planck suggested the establishment of Δ • “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.”**Planck Precision Limits**Quantum-scale granulation of reality Mass Length Time Area Volume Density Any measure Δ Δ**Planck Infinitesimals** L = lpl = (hG/c 3)1/2 = 10-33 cm m = mpl = (hc/G)1/2 = 10-5 g t = tpl= (hG/c 5)1/2 = 10-43 s**Abraham Robinson, Mathematician**• 1918 – 1974 • developed nonstandard analysis • a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics.**Smallest Measurable Length **South Carolina is to a Proton As a Proton is to a Planck length **The Quantum Limit**is the limit of measurability. It is the quantum limitof X in the differential quotient of Calculus.**Limited Real Precision**If real numbers are for measuring, And measuring precision is limited by quantum mechanics, Then measurable real numbers have limited precision.**A Lower Limit to Measurable Precision**L= 10-35 m The “infinitesimal”**An Old Paradox Revisited**• 1.999… = 1 + 9 * .111… • 1.999… = 1+ 9 * 1/9 • 1.999… = 1 + 1 • So 1.999… = 2 • But at the quantum edge, • 2 – 1.999… = Δ≠ 0 • So 2 ≠ 1.999… 1.9999999999999999999999999999999999999999999999999999999**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. 2 1**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.**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.**Reality-Math Point-Density Resolved**• Rounding b to the nearest Δ-integer shows that a:b is many-to-one, not 1-to-1**Pythagorus**• Good Old Pythagorus • c2 = a2 + b2 • True for all right triangles • then and now and forever • Maybe**Pythagorean Failures**• The hypotenuse of a quantum-scale isosceles right triangle, being aΔ – integer, cannot be irrational. • Three cases pertain.**Quantum Pythagorus Case 1**• The hypotenuse is a truncatedΔ– integer in a discontinuous triangle. • 9-9-12.729… • 9-9-12**Quantum Pythagorus Case 2**• The hypotenuse is a rounded-upΔ– integer in a continuous triangle with overlap. • 9-9-12.729… • 9-9-13**Quantum Pythagorus Case 3**• The triangle is continuous, • But the longest side is no hypotenuse because the triangle is not exactly right-angled.**Quantum Pythagorean Triples**• 3-4-5 • 5-12-13 • Is there a minimal angle? • 7-24-25?**Quantum Geometry is Different**• A = ½ BH • H = 2A / B • A = 15 balls • B = 5 balls • But H ≠ 6balls**Geometry at the Quantum Edge of Reality**• Circles, when pressed against each other • Become hexagons**There are Three Regular Tesselations of the Plane**• Nature chooses the hexagon**Natural Angles and Forms**• 60º • Equilateral triangles • No right triangles at the quantum edge**Quantum Angles**• Straight lines intersect at fixed angles of 60º and 120º**Quantum Hexagonal Grid**• Cartesian coordinates can translate into quantum hexagon sites**What is a Quantum Circle?**• A quantum circle is a hexagon**Quantum Circles**• Not all circumferences exist • Not all diameters exist • Not all “points” are equidistant from the center**Quantum Continuity**• Face-sharing may define continuity at the quantum edge • But it fails as a function.**Quantum Discontinuity**• Greater slopes cause discontinuity at the quantum edge • Only linear functions are continuous at the quantum edge**Integration is Discrete**• Quantum Integration is discrete • The integral is a Δ-sum • Discontinuous functions are integrable.**Quantum 3-D Structures**• What models will be useful in examining geometry at the quantum edge?