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The Unique Infinity of 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

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the unique infinity of the denumerable reals

The Unique Infinityof the Denumerable Reals

Mathematics on the Edge of Quantum Reality

dr brian l crissey
Dr. Brian L. Crissey
  • Professor of Mathematics
  • North Greenville University, SC
  • Math/CS 1975
  • Johns Hopkins
my path
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

One of Mathematics’ Great Traditions

12 / 4

= 3

= 0

today s intent
Today’s Intent

א0א1א2א3 …

To Simplify

Transfinite Mathematics

Down to…

{ φ } … the empty set

chart of numbers
Chart of Numbers

Potentially Infinite Precision

Finite Precision







infinite periodic precision
Infinite Periodic Precision
  • Periodic Reals have infinitely long decimal expansions
  • Example (1/7)10
    • 0.142857142857142857142857…
  • Where do they fit?
repeating expansions
Repeating Expansions


Potentially Infinite Precision

Finite Precision







eliminating infinite periodic precision
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
Revised Chart of Numbers


Potentially Infinite Precision

Finite Precision







are irrationals even real
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
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 disciplines in the 20 th century
Influential Disciplinesin the 20th Century

Computer Science




Has Math Integratedthe New Knowledge?

mathematical minds from the last century
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
From Quantum Physics

Everything is energy

Matter is perception of concentrated energy

Particle detector limit

Smallest “particle”




quantum geometry
Quantum Geometry

A Quantum point occupies a non-zero volume

Many implications

A quantum “point”




natural units
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.”
planck precision limits
Planck Precision Limits

Quantum-scale granulation of reality







Any measure



planck infinitesimals
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
Abraham Robinson, Mathematician
  • 1918 – 1974
  • developed nonstandard analysis
  • a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics.
smallest measurable length
Smallest Measurable Length 

South Carolina

is to a Proton

As a Proton is to a Planck length 

the quantum limit
The Quantum Limit

is the limit of measurability.

It is the quantum limitof X in the differential quotient of Calculus.

limited real precision
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
A Lower Limit to Measurable Precision

L= 10-35 m

The “infinitesimal”

an old paradox revisited
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…


classical 2 1 point paradox
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.



reality math 2 1 paradox revisited
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
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
Reality-Math Point-Density Resolved
  • Rounding b to the nearest Δ-integer shows that a:b is many-to-one, not 1-to-1
  • Good Old Pythagorus
  • c2 = a2 + b2
  • True for all right triangles
  • then and now and forever
  • Maybe
pythagorean failures
Pythagorean Failures
  • The hypotenuse of a quantum-scale isosceles right triangle, being aΔ – integer, cannot be irrational.
  • Three cases pertain.
quantum pythagorus case 1
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
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
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
Quantum Pythagorean Triples
  • 3-4-5
  • 5-12-13
  • Is there a minimal angle?
  • 7-24-25?
quantum geometry is different
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
Geometry at the Quantum Edge of Reality
  • Circles, when pressed against each other
  • Become hexagons
natural angles and forms
Natural Angles and Forms
  • 60º
  • Equilateral triangles
  • No right triangles at the quantum edge
quantum angles
Quantum Angles
  • Straight lines intersect at fixed angles of 60º and 120º
quantum hexagonal grid
Quantum Hexagonal Grid
  • Cartesian coordinates can translate into quantum hexagon sites
what is a quantum circle
What is a Quantum Circle?
  • A quantum circle is a hexagon
quantum circles
Quantum Circles
  • Not all circumferences exist
  • Not all diameters exist
  • Not all “points” are equidistant from the center
quantum continuity
Quantum Continuity
  • Face-sharing may define continuity at the quantum edge
  • But it fails as a function.
quantum discontinuity
Quantum Discontinuity
  • Greater slopes cause discontinuity at the quantum edge
  • Only linear functions are continuous at the quantum edge
integration is discrete
Integration is Discrete
  • Quantum Integration is discrete
  • The integral is a Δ-sum
  • Discontinuous functions are integrable.
quantum 3 d structures
Quantum 3-D Structures
  • What models will be useful in examining geometry at the quantum edge?
3 d quantum geometry
3-D Quantum Geometry
  • How do 3-D quanta arrange themselves naturally?
quantum tesselation
Quantum Tesselation
  • Spheres press together into 3-D tesselations.
a real partition
A Real Partition

Speculative reals may have infinite precision but are not computable

Measurablereals have finite precisionand are denumerable

Measurable Speculative

The Real Numbers

measurable vs speculative
Measurable vs. Speculative

The computation of √2 as a measureis truncated by Planck limits

√2 has infinite precisionbut never terminates..



√2 * √2 returns no value, as the process never terminates.

R = Rm U RS


redefining functions
Redefining Functions
  • A real function must return a result
  • This is not a function :
    • Y(X) = { 1, if x is rational -1, if x is irrational }
    • Y( P εRS) will not terminate
  • A function defined on Δ-integers, will always return a Δ-integer .
  • Cardinality (Z) = Cardinality (Rm) = ∞
but what about the speculative reals
But What About the Speculative Reals

Surely they are not denumerable



R = Rm U RS



Like √2 ε Rs


Never deliver a usable result


They truncate to a rational approximation εRm

surely pi is irrational
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

Pi: is a ratio of a two measurable reals

  • Measurable reals are Δ- integers
  • So pi is rational
the best estimate of pi
The Best Estimate of Pi

Would be the measure of the greatest knowable circle

Divided by the measure of its diameter

what about cantor
What About Cantor?
  • Is his work valid?
  • If not, what are the implications?
georg cantor a sketch
Georg Cantor: A Sketch
  • b. 1845 in St. Petersburg
  • 1856 Moved to Germany
  • 1867 Ph.D. in Number Theory, University of Berlin
  • Professor, University of Halle
  • In and out of mental hospitals all his life
  • 1918 died in a sanatorium
cantor s controversies
Cantor’s Controversies
  • Some Infinities are larger
  • Maybe
  • Infinities can be completed
  • Maybe
  • Cardinalities can be operated upon
  • Maybe
discomfort with actual infinities
Discomfort with Actual Infinities


384 BC -322 BC

  • Greek Philosopher
  • "The concept of actual infinity is internally contradictory"

“Infinitum actu non datur”


discomfort with actual infinities1
Discomfort with Actual Infinities

“There is no actual infinity- Cantorians forgot that and fell into contradiction...”

Henri Poincaré


  • Philosopher and Mathematician
  • Said that Cantor's work was a disease from which mathematics would eventually recover
discomfort with actual infinities2
Discomfort with Actual Infinities

Ludwig Wittgenstein


  • 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”

discomfort with cantor
Discomfort with Cantor

Alexander Alexandrovich Zenkin


  • “The third crisis in the foundations of mathematics was Georg Cantor’s cheeky attempt to actualize the Infinite.”
discomfort with cantor1
Discomfort with Cantor

Cantor’s theory was “a pathological incident in the history of mathematics from which future generations will be horrified.”

L.E.J. Brouwer


  • Dutch mathematician and philosopher
  • Founder of modern topology
  • Attempted to reconstruct Cantorian set theory
cantor s diagonal
Cantor’s Diagonal
  • Enumerate the reals
  • Output a non-denumerable real
  • Conclusion:
    • Reals are not denumerable
    • So Cardinality(R) > Cardinality(Z)
  • But Cantor produced a nonterminal output string, not a nondenumerable real
re examining cantor s diagonal proof
Re-examining Cantor’s Diagonal Proof
  • Cross-products of denumerable sets are denumerable
denumerable sets
Denumerable sets





















All men are created equal…

When in the course of human events…

  • Integers
  •  - Reals
  • Input Strings
  • Characters
  • Words
  • Sentences
  • Paragraphs
  • Procedures
input driven procedures
Input-Driven Procedures
  • are denumerable
  • Procedures are denumerable
  • Inputstringsaredenumerable
denumerating cantor
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;

Somewhere in the list of all possible procedures is Cantor’s procedure to generate a non-denumerable real

cantor s failed diagonal argument
Cantor’s Failed Diagonal Argument
  • Cantor’s non-enumerated real
  • Is just a process output
  • Matched digit by digit by the output of the correct enumerated procedure
  • There is no non-enumerated real



if cantor s wrong

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



According to truth tables

False implies anything is true

So if Cantor was wrong, we have falsely implied some conclusions

the continuum hypothesis
The Continuum Hypothesis
  • Hilbert 1900
  • First of 23 great Unanswered Math Questions
  • “Does there exist a cardinal between א0 & c?”

λbetween א0 and c

א0 ≤ λ ≤ c ?

implication 6
Implication 6
  • There is no cardinal between א0and c because they are equal.
david hilbert
David Hilbert
  • “No one shall drive us from the paradise Cantor created for us.”
driven from paradise

Driven from Paradise?

Is the Cantorian Church of PolyInfinitism in need of reform?

the theses


The ¯ Theses

There is but one infinity

Reals are denumerable

א0 = א1 = א2 = א3 … = ∞

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?

the kicked the bucket list

The “Kicked the Bucket” List

There are infinities of infinities

Reals are not denumerable

א0 < א1 < א2 < א3 …

Cardinality(R) = c = 2א0> א0 = C(Z)

Universality of Pythagorean Theorem


Transfinite Mathematics

Axiomatic Set Theory…

  • We have graduated into
    • The Quantum Mathematical Universe
  • Many things may change
the great circle
The GreatCircle
  • Math and Physics
  • Computer Science
  • CS changed too quickly
  • Math seemed stable
  • Now I’m not so sure.
  • Perhaps I’ll head back to CS
    • Where things don’t change so much…


  • א0