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

To Simplify

Transfinite Mathematics

Down to…

{ φ } … the empty set

0א 1א 2א3א

…

RATIONALS

Chart of Numbers

INTEGERS

Finite PrecisionPotentially Infinite Precision

21

21/6irrationals

REALS

Infinite Periodic Precision Periodic Reals have infinitely long

decimal expansions Example (1/7)10

– 0.142857142857142857142857… Where do they fit?

RATIONALS

Repeating Expansions

INTEGERS

Finite PrecisionPotentially Infinite Precision

21

21/6irrationals

REALS

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.

RATIONALS

Revised Chart of Numbers

INTEGERS

Finite PrecisionPotentially Infinite Precision

21

21/6

irrationals

REALS

Are Irrationals Even Real?

Leopold Kronecker1823 - 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.”

Irrationals Never Reach The Real Number Line

Asymptotic Approach of Square Root of 2 to the RNL

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

Approximations to Square Root of 2

Pct

of

Journ

ey C

om

ple

ted

J ourney

What is a Real Number?

Solomon Feferman1928 – present

Mathematician and philosopher at Stanford University

Author of – In the Light of Logic

Reals are those numbers intended for measuring.

Influential Disciplinesin the 20th Century

Physics Computer Science

QuantumTheory

Computability

Has Math Integratedthe New Knowledge?

Mathematical Mindsfrom the Last Century

PhysicsQuantum

TheoryAnd the Limits

of Measurability Computer

ScienceComputabilityAnd

Enumeration Time to

Upgrade?

Alan TuringMax Planck

From Quantum Physics

Everything is energy Matter is perception of

concentrated energy

“Particles”

“Waves”

Particle detector limit Smallest “particle”

Δ

Quantum Geometry

A Quantum point occupies a non-zero volume

Many implications

“Particles”

“Waves”

A quantum “point”

Δ

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

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

As a Proton is to a Planck length

is to a Proton

The Quantum Limit

is the limit of measurability.

It is the quantum limit of 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

DL = 10-35 mThe

“infinitesimal”

The Measurable Universe is Granular

V

Implication 1

Two real measures that differ by less than are indistinguishable in our reality.

If |r1 – r2| < D then r1 = r2

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

b a

a1

a R(b)1 12 13 24 25 26 27 28 39 3

10 311 312 313 314 415 416 417 418 419 420 421 422 523 524 525 5

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 ≠ 6

balls

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

Circumference Diameter Pi?1 1 1.06 3 2.012 5 2.4

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

Measurable reals have finite precisionand are denumerable

Measurable Speculative

The Real Numbers

Speculative reals may

have infinite precision but

are not computable

Measurable vs. SpeculativeThe computation of √2 as a measure is truncated by Planck limits

R = Rm U RS

√2 has infinite precision

but never terminates..

1.4142135623730950488016887242097…

RsRm

√2 * √2 returns no

value, as the process never

terminates.

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 ε RS) will not terminate A function defined on Δ-integers,

will always return a Δ-integer .

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

┘

AE

Implication 3

If cardinality (Z) = א0, then

cardinality (Rm) = א0

Simplification

Cardinality (Z) = Cardinality (Rm) = ∞

But What About the Speculative Reals

Surely they are not denumerable

R = Rm U RS

1.4142135623730950488016887242097…

RsRm

Irrationals

Like √2 ε Rs

– 1.41421356237309504880168872… Never deliver a usable result Or

– They truncate to a rational approximation ε Rm

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

Estimating Rational Pi

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

Aristotle384 BC -322 BC

Greek Philosopher

"The concept of actual infinity is internally contradictory"

“Infinitum actu non datur”

-Aristotle

Discomfort with Actual Infinities

Henri Poincaré1854-1912

Philosopher and Mathematician

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...”

Discomfort with Actual Infinities

Ludwig Wittgenstein1889-1951

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

L.E.J. Brouwer1881-1966

Dutch mathematician and philosopher

Founder of modern topology

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.”

Cantor’s Diagonal Enumerate the reals Output a

non-denumerable real Conclusion:

– Reals are not denumerable– So Cardinality(R) > Cardinality(Z)

But Cantor produceda nonterminal output string, not a nondenumerable real

Re-examining Cantor’s Diagonal Proof Cross-products of denumerable

sets are denumerable

Denumerable sets

Integers - Reals Input Strings Characters Words Sentences Paragraphs Procedures

1234…101112…99…999…abc…aaabac…zz…zzz…alphabeta…omega…All men are created equal…When in the course of human events…

Input-Driven Procedures

Procedures are denumerable

Inputstringsaredenumerable

are denumerable

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

CANTOR

2.32514…

Implication 5

Cardinality (Z) = א0

=

cardinality (Rm

) =

cardinality (Rs) = ∞

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

Implications

According to truth tablesFalse implies anything is trueSo 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 0א & c?”

λ between 0א and c

0א ≤ λ ≤ c ?

Implication 6

The Continuum Hypothesis can be confirmed.

א0

= c = ∞

There is no cardinal between 0א and c because they are equal.

David Hilbert

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

Driven from Paradise?

Is the Cantorian Church of PolyInfinitism in need of reform?

The ¯ Theses

There is but one infinityReals are denumerable 3 = א2 = א1 = א0א … = ∞

Cardinality(R) = c = ∞ = C(Z)There are no right triangles at the

Quantum EdgeGeometry changes at the Quantum

EdgeWhat else has kicked the bucket?

.99

The “Kicked the Bucket” List

There are infinities of infinitiesReals are not denumerable 3 > א2 > א1 > א0א …Cardinality(R) = c = 2

א0

0< א = C(Z)Universality of Pythagorean TheoremMetamathematicsTransfinite MathematicsAxiomatic 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…

A New Beginning