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### The Unique Infinityof the Denumerable Reals

### If Cantor’s Wrong…

### Implications

### Driven from Paradise?

### The ¯ Theses

### The “Kicked the Bucket” List

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

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.

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.

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?

3-D Quantum Geometry

- How do 3-D quanta arrange themselves naturally?

Quantum Tesselation

- Spheres press together into 3-D tesselations.

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

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

√2 has infinite precisionbut never terminates..

Rm

Rs

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

R = Rm U RS

1.4142135623730950488016887242097…

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 .

Simplification

- Cardinality (Z) = Cardinality (Rm) = ∞

But What About the Speculative Reals

Surely they are not denumerable

Rm

Rs

R = Rm U RS

1.4142135623730950488016887242097…

Irrationals

Like √2 ε Rs

1.41421356237309504880168872…

Never deliver a usable result

Or

They truncate to a rational approximation εRm

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

Would be the measure of the greatest knowable circle

Divided by the measure of its diameter

What About Cantor?

- Is his work valid?
- If not, what are the implications?

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

- Some Infinities are larger
- Maybe
- Infinities can be completed
- Maybe
- Cardinalities can be operated upon
- Maybe

Discomfort with Actual Infinities

Aristotle

384 BC -322 BC

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

“Infinitum actu non datur”

-Aristotle

Discomfort with Actual Infinities

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

Henri Poincaré

1854-1912

- Philosopher and Mathematician
- Said that Cantor's work was a disease from which mathematics would eventually recover

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”

Discomfort with Cantor

Alexander Alexandrovich Zenkin

1937-2006

- “The third crisis in the foundations of mathematics was Georg Cantor’s cheeky attempt to actualize the Infinite.”

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

1881-1966

- Dutch mathematician and philosopher
- Founder of modern topology
- Attempted to reconstruct Cantorian set theory

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

- Cross-products of denumerable sets are denumerable

Denumerable sets

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…

- Integers
- - Reals
- Input Strings
- Characters
- Words
- Sentences
- Paragraphs
- Procedures

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 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

2.32514…

CANTOR

“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

- 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

- There is no cardinal between א0and c because they are equal.

David Hilbert

- “No one shall drive us from the paradise Cantor created for us.”

Is the Cantorian Church of PolyInfinitism in need of reform?

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?

There are infinities of infinities

Reals are not denumerable

א0 < א1 < א2 < א3 …

Cardinality(R) = c = 2א0> א0 = C(Z)

Universality of Pythagorean Theorem

Metamathematics

Transfinite Mathematics

Axiomatic Set Theory…

Conclusion

- We have graduated into
- The Quantum Mathematical Universe
- Many things may change

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…

א5

- א0

א5

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