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Mathematics 4390 Senior Seminar On Infinities

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Dr. Brian L. Crissey Chief Rabble Rouser Chief Rabble Rouser Playing the role of Playing the role of –Martin Luther

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Co-conspirators Andrew Blalock Andrew Blalock Anastasia Bridner Anastasia Bridner Jessica Farmer Jessica Farmer Karla Kirby Karla Kirby Monica McAbee Monica McAbee Eric Moeller Eric Moeller Delecta Rollins Delecta Rollins Christopher Tate Christopher Tate Contessa Wright Contessa Wright

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Delecta Rollins Who are we? Who are we? What are we tying to accomplish? What are we tying to accomplish?

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Carl Friedrich Gauss “Mathematics is the queen of all sciences, and the Theory of Numbers is the queen of mathematics." “Mathematics is the queen of all sciences, and the Theory of Numbers is the queen of mathematics."

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Georg Cantor Cardinality Cardinality Multiple Infinities Multiple Infinities Insane Insane –Or a genius?

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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.”

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Driven from Paradise Reforming the Cantorian Church of PolyInfinitism

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Andrew Blalock Male Scholar Athlete of the Year (2006) Male Scholar Athlete of the Year (2006) Philosophy Philosophy Georg Cantor Georg Cantor

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New Ideas How have they been received in the past? How have they been received in the past?

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Ptolomy’s Universe Earth- centered Earth- centered Stars fixed Stars fixed

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Copernicus’s Universe Heliocentric Heliocentric Kept silent Kept silent Died before reaction Died before reaction

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Bruno’s Universe Infinite Infinite No center No center Outspoken Outspoken Burned at the stake Burned at the stake

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What Will Be Our Fate? Ignored Ignored Scoffed at Scoffed at Castigated Castigated Accepted Accepted

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Philosophies Plato Plato Platonism Platonism Monism Monism One truth One truth

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Philosophies Pluralism Pluralism Many Truths Many Truths No absolute truth No absolute truth

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Our Philosophy Pragmatism Pragmatism Earth-relevant Earth-relevant What makes sense What makes sense On a finite planet On a finite planet In a Quantum Universe In a Quantum Universe

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Gottfried Wilhelm Leibnitz “Drawing is a very useful tool against the uncertainty of words.” “Drawing is a very useful tool against the uncertainty of words.” So we will be as visual as possible. So we will be as visual as possible.

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Georg Cantor The man The man The mathematician The mathematician His contributions His contributions His controversies His controversies

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Georg Cantor: His Life Born in 1845 in St. Petersburg, Russia Born in 1845 in St. Petersburg, Russia Moved to Germany in 1856 Moved to Germany in 1856 In 1867, he received his Ph.D. in Number Theory from the University of Berlin. In 1867, he received his Ph.D. in Number Theory from the University of Berlin. He became a professor at the University of Halle where he remained for the rest of his career. He became a professor at the University of Halle where he remained for the rest of his career.

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Georg Cantor: His Life In 1874, he was married to Vally Guttman and had 6 children. In 1874, he was married to Vally Guttman and had 6 children. In 1879, he was promoted to full professorship, an impressive achievement. In 1879, he was promoted to full professorship, an impressive achievement.

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Georg Cantor: His Struggle Cantor’s desire was to become a professor at a more prestigious institution (Berlin) but Kronecker, Cantor’s former teacher and chair of mathematics at Berlin would not allow it. This conflict led to many nervous breakdowns. Cantor’s desire was to become a professor at a more prestigious institution (Berlin) but Kronecker, Cantor’s former teacher and chair of mathematics at Berlin would not allow it. This conflict led to many nervous breakdowns. In 1899, Cantor’s youngest son’s death sent him into a chronic depression, keeping him hospitalized for much of his later life. In 1899, Cantor’s youngest son’s death sent him into a chronic depression, keeping him hospitalized for much of his later life. Cantor died in a mental institution in 1918. Cantor died in a mental institution in 1918.

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Georg Cantor: His Accomplishment Cantor proposed the idea that the real numbers have a greater cardinality than the integers. Cantor proposed the idea that the real numbers have a greater cardinality than the integers. Cantor determined the smallest transfinite number, א, represents the cardinality of the integers because they are denumerable, while the real numbers have a cardinality of C, a “higher” transfinite because they are not denumerable. Cantor determined the smallest transfinite number, א, represents the cardinality of the integers because they are denumerable, while the real numbers have a cardinality of C, a “higher” transfinite because they are not denumerable. Cantor is also responsible for the establishment of Set Theory as a branch of mathematics. Cantor is also responsible for the establishment of Set Theory as a branch of mathematics.

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Contessa Wright Recreational Specialist Recreational Specialist Member of American Chemical Society Member of American Chemical Society “Many small people in many small places who do many small things, can alter the face of the world.” –Berlin Wall “Many small people in many small places who do many small things, can alter the face of the world.” –Berlin Wall Part 1: Part 1: Cantor’s Diagonal Argument Cantor’s Diagonal Argument

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Cantor’s Diagonal Argument Enumerating the reals Enumerating the reals The non-enumerated real The non-enumerated real The contradiction The contradiction The conclusion The conclusion Completed Infinities Completed Infinities Multiple infinities Multiple infinities

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Jessica Farmer Mathematics Major Mathematics Major Spanish Minor Graduating in December 2007 Graduating in December 2007 Literature Review Literature Review

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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.” Supporter of alternate theories to Cantor’s theory

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Discomfort with Actual Infinities Aristotle 384 BC -322 BC Greek Philosopher Greek Philosopher Distinguished between 2 types of infinity: Distinguished between 2 types of infinity: - potential - actual "The concept of actual infinity is internally contradictory" "The concept of actual infinity is internally contradictory" “Infinitum actu non datur” -Aristotle

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Discomfort with Actual Infinities Henri Poincaré 1854-1912 Philosopher and Mathematician Philosopher and Mathematician Claimed there is no actual Infinity Claimed there is no actual Infinity 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...”

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Discomfort with Actual Infinities … Poincaré continued 2 Classifications predicative- DO NOT change with introduction of new elements impredicative- DO change with introduction of new elements impredicative- DO change with introduction of new elements Poincaré argues that Cantor’s proof, which is based on the assumption of a real infinity, is impredicative.

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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.”

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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”

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Discomfort with Cantor Leopold Kronecker 1823 - 1891 Cantor’s Mentor 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.”

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Discomfort with Actual Infinities Solomon Feferman 1928 – present Mathematician and philosopher at Stanford University Mathematician and philosopher at Stanford University Author of In the Light Author of In the Light of Logic Agrees that Cantor’s Agrees that Cantor’s theory is not necessary for mathematics “The actual infinity is a self- contradictory notion, and its usage in mathematics is inadmissible.”

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Eric Moeller Winner of “Math Major w/ Best Style” Winner of “Math Major w/ Best Style” Loves Math as much as the 80’s Loves Math as much as the 80’s Reality Extrema Reality Extrema

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Reality Mathematics for a Finite Planet The Religious War The Religious War –Kronecker vs. Cantor Reality Extrema Reality Extrema The Planck limits The Planck limits Infinite Precision Infinite Precision

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Zeno’s Dichotomy The infinitely large The infinitely large The infinitely small The infinitely small

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The Religious War Cantor: “Infinitely divisible numbers lie between any two whole numbers." Kronecker: “My dear Lord God made all the integers. Everything else is the work of Man."

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Time Limits Length

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Planck Limits Quantum-scale limits Quantum-scale limits –Mass –Length –Area –Volume –Time

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Smallest Meaningful Length New Jersey is to a Proton New Jersey is to a Proton As a Proton is to a Planck length As a Proton is to a Planck length

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Delta Infinitesimal The X of integral calculus is the quantum limit of (X2 - X1) The X of integral calculus is the quantum limit of (X2 - X1) is the legendary infinitesimal is the legendary infinitesimal X1X2 X

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The Smallest Meaningful Length is the limit of measurability. It is the limit of X in the differential quotient of Calculus.

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God’s Unit Size is the basic unit size of the Universe. is the legendary infinitesimal. Every meaningful number is an integer, measured in s. Integers are denumerable. Real numbers are integers, so they too are denumerable.

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The Question of Infinite Precision What to do with real numbers whose precision is infinite? Irrationals like square root of 2? –1–1.414213562… Periodic decimal expansions like 1/7? –0–0.142857142857142857142857…

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Asymptotic Approach of Irrationals

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Karla Kirby “Class Chaplain” Mathematics Major Spanish Minor Graduating December 2007 Restructuring the Reals The Reality Number Line

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Current Mathematics One Real Number Line –I–I–I–Includes Rational Numbers –I–I–I–Includes Irrational Numbers What is the problem?

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The Problem Irrational Numbers Irrational Numbers –Pi 3.14159 26535 89793 23846 26433 83279 50288 4197… 3.14159 26535 89793 23846 26433 83279 50288 4197… –Euler’s Number 2.71828 18284 59045 23536 02874 71352 66249 7757… 2.71828 18284 59045 23536 02874 71352 66249 7757… –Pythagoras’s Constant 1.41421 35623 73095 04880 16887 24209 69807 85696… 1.41421 35623 73095 04880 16887 24209 69807 85696… –All Non-Repeating Infinite Decimals

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The Problem Where is the “…” on our current Real Number Line?

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Redefining the Reals Reality Numbers –C–C–C–Completed Reality Numbers –P–P–P–Process Output Strings

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Completed Reality Numbers –D–D–D–Definition Includes all rational numbers Represented by a positive or negative integer, a decimal point, and a second integer

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Completed Reality Numbers –P–P–P–Precision Fractional part - safely rounded to 36 digits (Hr) with no loss of verifiable meaning in reality

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Completed Reality Numbers –S–S–S–Scale Precision in reality is 36 decimal digits. Precision for Completed Reality Numbers is 36 decimal digits.

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Precision Output Strings –D–D–D–Definition Includes all irrational numbers Generated by non-terminal processes Includes all ideas that generate infinitely long sequences of digits

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All Precision Output Strings become Reality Numbers Precision Output Strings –P–P–P–Precision and Scale Currently – an infinite number of decimal digits ALL DECIMALS TO PRECISION OF 36 DECIMAL PLACES

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Redefining the Reals Reality Number Line –A–A–A–An updated Real Number Line –P–P–P–Precision – 36 decimal digits –A–A–A–Accuracy – No loss of verifiable meaning in reality

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Christopher Tate NGU Baseball NGU Baseball Enumeration of rational numbers Enumeration of rational numbers Infinite Precision Resolution Infinite Precision Resolution Enumeration of processes Enumeration of processes Results Results

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Enumerating the Rationals Example Example Cross-products of denumerable sets are denumerable Cross-products of denumerable sets are denumerable

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Eliminating Infinite Periodic Precision Periodic Reals have infinitely long decimal expansions Example (1/7)10 ––0––0.142857142857142857142857… Eliminate the issue by changing the base to the fraction’s denominator (1/7)10 = (0.1) 7 Radix is a presentation issue, not an existence issue.

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Enumerating Text Strings Letters are denumerable Letters are denumerable As are words As are words As are sentences As are sentences

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Enumerating Processes Processes are denumerable. Processes are denumerable. One of these processes continually outputs Cantor’s Non- denumerable real. One of these processes continually outputs Cantor’s Non- denumerable real.

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Enumerating Cantor’s Non-Denumerable Real

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Results Periodic Rationals can be converted into reality numbers Periodic Rationals can be converted into reality numbers –By radix conversion. Processes are denumerable. Processes are denumerable. Cantor’s Non-denumerable Real will be produced by one of the enumerated processes. Cantor’s Non-denumerable Real will be produced by one of the enumerated processes. Reals are denumerable. Reals are denumerable.

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Contessa Wright Part 2: Part 2: The Dismissal of Cantor’s Diagonal Argument The Dismissal of Cantor’s Diagonal Argument Goodbye Contradictions Goodbye Contradictions

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The Need to Redefine the Real Numbers An infinitely long digit expansion cannot be enumerated, An infinitely long digit expansion cannot be enumerated, –because it will never terminate. There is no meaningful precision more precise than . There is no meaningful precision more precise than . Yet classical real numbers include irrationals, with infinitely long digit expansions. Yet classical real numbers include irrationals, with infinitely long digit expansions. There is a need to redefine the real numbers. There is a need to redefine the real numbers.

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Cantor’s Failed Diagonal Argument The non-enumerated real The non-enumerated real Is just a process output Is just a process output Enumerated with the process of ratio-of- integers p/q, where p ^ q are ε Z and q ≠ 0 Enumerated with the process of ratio-of- integers p/q, where p ^ q are ε Z and q ≠ 0

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DeCantorizing the Argument There is no contradiction There is but one infinity There is no completed infinity 1.999... = 1 +.999… 1.999… = 1 + 9 *.111… 1.999… = 1+ 9 * 1/9 1.999… = 1 + 1 1.999… = 2 X

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Anastasia Bridner Honors Graduate Honors Graduate Recipient of the Excellence in Mathematics Award Recipient of the Excellence in Mathematics Award Implications Implications

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Implications “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 ‘Babel-2’ of these sciences fall to pieces as a house of cards.” Alexander Zenkin

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Implications Quantum Geometry Quantum Geometry Infinitesimal Polygons Infinitesimal Polygons Resolving Paradoxes Resolving Paradoxes Redefining Functions Redefining Functions Redefining Continuity Redefining Continuity Exact Integration Exact Integration

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Quantum Geometry Hypotenuse of a right triangle with two sides of length 1 is not irrational. Hypotenuse of a right triangle with two sides of length 1 is not irrational. It is a Reality Number rounded to a precision sufficient for a task. It is a Reality Number rounded to a precision sufficient for a task. Pythagorean Theorem produces approximations, not irrationals. Pythagorean Theorem produces approximations, not irrationals.

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Quantum Pythagorus The hypotenuse of a right triangle with short sides of length 1 unit should be 1.41421… units, The hypotenuse of a right triangle with short sides of length 1 unit should be 1.41421… units, which is not a choice when the units are infinitesimals which is not a choice when the units are infinitesimals

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Infinitesimal Squares As the sides of a square approach the Planck limit, As the sides of a square approach the Planck limit, The least square appears The least square appears

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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

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Reality Math 2:1 Paradox Resolved There are twice as many infinitesimals in a line segment of length 2 as there are in a line segment of length 1. There are twice as many infinitesimals in a line segment of length 2 as there are in a line segment of length 1.

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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.

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Reality-Math Point Density Resolved Rounding b to the nearest infinitesimal on the Reality Number Line shows that the relationship is many-to- one, not 1-to-1

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Redefining Functions A function must return a result A function must return a result Not a function : Not a function : –Y(X) = { 1, if x is rational -1, if x is irrational } –Y( ) will not terminate A function of reality numbers, defined at a reality number, will always return a reality number. A function of reality numbers, defined at a reality number, will always return a reality number.

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Redefining Continuity Slopes |s| <=1 are continuous in quantum reality Slopes |s| <=1 are continuous in quantum reality A Continuity Delta spreads to the right A Continuity Delta spreads to the right

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Redefining Discontinuity Slopes |s| >1 are discontinuous in quantum reality Slopes |s| >1 are discontinuous in quantum reality

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Integration is Exact Integration is exact in quantum reality Integration is exact in quantum reality

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Discontinous Integration Even Discontinuous Integration is exact in quantum reality Even Discontinuous Integration is exact in quantum reality

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Monica McAbee Physics Facilitator Group Leader Physics Facilitator Group Leader Science Division Work Study Science Division Work Study Summary Summary Conclusions Conclusions

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Summary Periodic numbers Periodic numbers –Can be transformed into Reality Numbers Numbers with infinite expansions Numbers with infinite expansions –Are not reality numbers –But outputs from processes Processes are denumerable Processes are denumerable Reality numbers are denumerable Reality numbers are denumerable

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Conclusion There is only one infinity There is only one infinity –Not an infinity of infinities Transfinite mathematics may be ignored Transfinite mathematics may be ignored We have graduated into We have graduated into –The Quantum Mathematical Universe

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A Final Thought… Ptolomy once contended that the Universe is Earth-centered, but he was discredited. Ptolomy once contended that the Universe is Earth-centered, but he was discredited.

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Now we notice that… The Knowable Universe that expands at the speed of light… The Knowable Universe that expands at the speed of light… Is Earth-Centered, Is Earth-Centered, …as Ptolomy once contended …as Ptolomy once contended Some things never change Some things never change

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The Beginning

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