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
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