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 But CS changed too quickly Math seemed stable Or so I thought

Simplification

One of Mathematics’ Great Traditions 12 / 4

Today’s Intent

To Simplify Transfinite Mathematics Down to… { φ } … the empty set א 0 א 1 א 2 א 3 …

Chart of Numbers

Potentially Infinite Precision Finite Precision REALS irrationals 21/6 RATIONALS INTEGERS 21

Infinite Periodic Precision

   Periodic Reals have infinitely long decimal expansions Example (1/7) 10 – 0.142857

142857 142857 142857… Where do they fit?

Repeating Expansions

Potentially Infinite Precision Finite Precision REALS irrationals 21/6 RATIONALS INTEGERS 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

Potentially Infinite Precision Finite Precision REALS irrationals 21/6 RATIONALS INTEGERS 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.”

Irrationals Never Reach The Real Number Line Asymptotic Approach of Square Root of 2 to the RNL

100.00% 90.00% 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 1.

1.

4 1.

41 1.

41 4 1.

41 42 1.

41 42 1 1.

41 42 13 1.

41 42 13 1.

5 41 42 13 1.

56 41 42 13 1.

56 41 2 42 13 1.

56 41 23 42 13 1.

56 41 23 42 7 13 1.

56 41 23 42 73 13 56 23 73 1

Approximations to Square Root of 2

Journey

What is a Real Number?

Solomon Feferman 1928 – present

Reals are those numbers intended for measuring.

 Mathematician and philosopher at Stanford University  Author of – In the Light of Logic

Influential Disciplines in the 20 th Century

Physics Quantum Theory Computability Computer Science Has Math Integrated the New Knowledge?

Mathematical Minds from 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 energy of concentrated Particle detector limit Smallest “particle” Δ “Particles” “Waves”

Quantum Geometry

  A Quantum point occupies a non-zero volume Many implications A quantum “point” Δ “Particles” “Waves”

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 = l pl = m = m pl ( hG/c 3 ) = (hc/G) 1/2 1/2 t = t pl = ( hG/c 5 ) 1/2 = 10 -33 cm = 10 -5 g = 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 limit of Calculus.

 X in the differential quotient of

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”

The Measurable Universe is Granular

 V

Implication 1

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

If |r1 – r2| <  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 to-1 Δ -integer shows that a:b is many-to-one, not 1 

b



a



a

 1 

a 1 2 7 8 9 3 4 5 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 R(b) 1 1 2 3 3 2 2 2 2 4 5 5 5 5 3 3 3 3 4 4 4 4 4 4 4

Pythagorus

     Good Old Pythagorus c 2 = a 2 + b 2 True for all right triangles then and now and forever Maybe

Pythagorean Failures

  The hypotenuse of a quantum-scale isosceles right triangle, being a Δ – integer , cannot be irrational.

Three cases pertain.

Quantum Pythagorus Case 1

  The hypotenuse is a truncated Δ – integer in a discontinuous triangle.

9-9 12.729…  9-9-12

Quantum Pythagorus Case 2

  The hypotenuse is a rounded-up Δ – 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

   60 º Equilateral triangles No right triangles at the quantum edge

Quantum Angles

 Straight lines intersect at fixed angles of 60 º and 120 º

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

1 1 6 12 3 5

Pi?

1.0

2.0

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 precision and are denumerable Measurable Speculative Speculative reals may have infinite precision but are not computable The Real Numbers

Measurable vs. Speculative

The computation of √ 2 as a measure is truncated by Planck limits

R m R s

R =

R m

U

R S

1.4142135623730950488016887242097… √ 2 has infinite precision but never terminates..

√ 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

R S

) 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.

E A

r i ε R ε Z m such that r = i * Δ And i = └ r/ Δ ┘

Implication 3

Integers are denumerable, so measurable reals are denumerable.

If cardinality (Z) = א 0 , then cardinality (R m ) = א 0

Simplification

Cardinality (Z) = C ardinality (R m ) =

∞

But What About the Speculative Reals

Surely they are not denumerable

R m R s

R =

R m

U

R S

1.4142135623730950488016887242097…

Irrationals

   Like √ 2 ε R s – 1.41421356237309504880168872… Never deliver a usable result Or – They truncate to a rational approximation ε R m

Surely Pi is Irrational?

 Pi: ratio of a circle’s circumference to its diameter  Circumference: measure  measure of a circle’s perimeter Diameter: The of a circle’s width

  Pi: is a two measurable reals Measurable reals are Δ ratio - integers of a  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 Aristotle

384 BC -322 BC “Infinitum actu non datur” -Aristotle Greek Philosopher  "The concept of actual infinity is internally contradictory"

Discomfort with Actual Infinities Henri Poincaré

1854-1912

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

 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

Cantor’s ideas of uncountable sets and different levels

of infinity are “a cancerous growth on the body of mathematics”  Rejected Cantor saying his argument “has no deductive content at all”

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. Brouwer 1881-1966  Dutch mathematician and philosopher

Cantor’s theory

was “a pathological incident in the history of mathematics from which future generations will be horrified.”  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

Integers  Reals Input Strings Characters Words Sentences Paragraphs Procedures 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…

Input-Driven Procedures are denumerable

  Procedures are denumerable Input strings are denumerable

Denumerating Cantor

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

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

Implication 5

Measurable reals are denumerable, and speculative reals are denumerable, so all reals are denumerable.

Cardinality (Z) = cardinality (R m א cardinality (Rs) = 0 ) = ∞ =

If Cantor’s Wrong…

“Cantor’s [diagonal] theorem is the cards 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 to pieces as a house of all the transfinite … sciences fall Alexander Zenkin

Implications

 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

א 0

& c?”

λ

between

א 0

and c

א 0 ≤ λ ≤

c ?

Implication 6

The Continuum Hypothesis can be confirmed.

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

and c

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?

 There is but one infinity  Reals are denumerable  א 0  = א 1 = א 2 = א 3 … = ∞ 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

 There are infinities of infinities  Reals are not denumerable  א 0  < א 1 < א 2 < א 3 … Cardinality(R) = c = 2 א 0 > א 0 = 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 Great Circle

      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