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Mathematics 4390
Senior SeminarOn Infinities
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Dr. Brian L. Crissey
Chief Rabble Rouser
Playing the role of– Martin Luther
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Co-conspirators
Andrew Blalock Anastasia Bridner Jessica Farmer Karla Kirby Monica McAbee Eric Moeller Delecta Rollins Christopher Tate Contessa Wright
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Delecta Rollins
Who are we? What are we
tying to accomplish?
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Carl Friedrich Gauss
“Mathematics is the queen of all sciences, and the Theory of Numbers is the queen of mathematics."
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Georg Cantor
Cardinality Multiple
Infinities Insane
– Or a genius?
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David Hilbert
“No one shall drive us from the paradise Cantor created for us.”
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Driven from Paradise
Reforming the Cantorian Church of PolyInfinitism
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Andrew Blalock
Male Scholar Athlete of the Year (2006)
Philosophy Georg Cantor
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New Ideas
How have they been received in the past?
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Ptolomy’s Universe
Earth-centered
Stars fixed
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Copernicus’s Universe
Heliocentric Kept silent Died before
reaction
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Bruno’s Universe
Infinite No center
Outspoken Burned at the
stake
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What Will Be Our Fate?
Ignored Scoffed at Castigated Accepted
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Philosophies
Plato Platonism Monism One truth
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Philosophies
Pluralism Many Truths No absolute
truth
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Our Philosophy
Pragmatism Earth-
relevant What makes
sense On a finite
planet In a
Quantum Universe
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Gottfried Wilhelm Leibnitz “Drawing is a very
useful tool against the uncertainty of words.”
So we will be as visual as possible.
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Georg Cantor
The man The
mathematician
His contributions
His controversies
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Georg Cantor: His Life
Born in 1845 in St. Petersburg, Russia
Moved to Germany in 1856 In 1867, he received his Ph.D. in
Number Theory from the University of Berlin.
He became a professor at the University of Halle where he remained for the rest of his career.
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Georg Cantor: His Life
In 1874, he was married to Vally Guttman and had 6 children.
In 1879, he was promoted to full professorship, an impressive achievement.
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Georg Cantor: His Struggle Cantor’s desire was to become a
professor at a more prestigious institution (Berlin) but Kronecker, Cantor’s former teacher and chair of mathematics at Berlin would not allow it. This conflict led to many nervous breakdowns.
In 1899, Cantor’s youngest son’s death sent him into a chronic depression, keeping him hospitalized for much of his later life.
Cantor died in a mental institution in 1918.
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Georg Cantor: His Accomplishment
Cantor proposed the idea that the real numbers have a greater cardinality than the integers.
Cantor determined the smallest transfinite number, א , represents the cardinality of the integers because they are denumerable, while the real numbers have a cardinality of C, a “higher” transfinite because they are not denumerable.
Cantor is also responsible for the establishment of Set Theory as a branch of mathematics.
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Contessa Wright
Recreational Specialist Member of American
Chemical Society
“Many small people in many small places who do many small things, can alter the face of the world.” –Berlin Wall
Part 1: Cantor’s Diagonal
Argument
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Cantor’s Diagonal Argument
Enumerating the reals
The non-enumerated real
The contradiction The conclusion Completed Infinities Multiple infinities
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Jessica Farmer
Mathematics MajorSpanish Minor
Graduating in December 2007
Literature Review
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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.”
Supporter of alternate theories to Cantor’s theory
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Discomfort with Actual Infinities
Aristotle384 BC -322 BC
Greek Philosopher
Distinguished between 2 types of infinity:- potential- actual
"The concept of actual infinity is internally contradictory"
“Infinitum actu non datur”
-Aristotle
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Discomfort with Actual Infinities
Henri Poincaré1854-1912
Philosopher and Mathematician
Claimed there is no actual Infinity
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...”
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Discomfort with Actual Infinities
… Poincaré continued
2 Classifications
predicative- DO NOT change with introduction of
new elements
impredicative- DO change with introduction of
new elements
Poincaré argues that Cantor’s proof, which is based on the assumption of a real infinity, is impredicative.
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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.”
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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”
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Discomfort with Cantor
Leopold Kronecker1823 - 1891
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.”
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Discomfort with Actual Infinities
Solomon Feferman1928 – present
Mathematician and philosopher at Stanford University
Author of In the Light of Logic
Agrees that Cantor’s theory is not necessary for mathematics
“The actual infinity is a
self-contradictory notion, and its usage in
mathematics is
inadmissible.”
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Eric Moeller
Winner of “Math Major w/ Best Style”
Loves Math as much as the 80’s
Reality Extrema
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Reality Mathematics for a Finite Planet
The Religious War– Kronecker vs. Cantor
Reality Extrema The Planck limits Infinite Precision
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Zeno’s Dichotomy
The infinitely large The infinitely small
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The Religious War
Cantor: “Infinitely divisible numbers lie between any two whole numbers."
Kronecker: “My dear Lord God made all the integers. Everything else is the work of Man."
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Time Limits Length
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Planck Limits
Quantum-scale limits– Mass– Length– Area– Volume– Time
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Smallest Meaningful Length New Jersey is to a Proton As a Proton is to a Planck length
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Delta Infinitesimal
The X of integral calculus is the quantum limit of (X2 - X1)
is the legendary infinitesimal
X1
X2X
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The Smallest Meaningful Length is the limit of
measurability. It is the limit of X in the differential quotient of Calculus.
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God’s Unit Size
is the basic unit size of the Universe.
is the legendary infinitesimal.
Every meaningful number is an integer, measured in s.
Integers are denumerable. Real numbers are integers,
so they too are denumerable.
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The Question of Infinite Precision What to do with real numbers
whose precision is infinite? Irrationals like square root of 2?
– 1.414213562… Periodic decimal expansions like
1/7?– 0.142857142857142857142857…
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Asymptotic Approach of Irrationals
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
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Karla Kirby
“Class Chaplain” Mathematics
Major Spanish Minor Graduating
December 2007 Restructuring
the Reals The Reality
Number Line
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Current Mathematics
One Real Number Line– Includes Rational Numbers– Includes Irrational Numbers
What is the problem?
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The Problem
Irrational Numbers– Pi
3.14159 26535 89793 23846 26433 83279 50288 4197…
– Euler’s Number 2.71828 18284 59045 23536 02874 71352 66249
7757…
– Pythagoras’s Constant 1.41421 35623 73095 04880 16887 24209 69807
85696… – All Non-Repeating Infinite Decimals
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The Problem
Where is the “…” on our current Real Number Line?
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Redefining the Reals
Reality Numbers– Completed Reality Numbers– Process Output Strings
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Redefining the Reals
Completed Reality Numbers– Definition
Includes all rational numbers Represented by a positive or negative
integer, a decimal point, and a second integer
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Redefining the Reals
Completed Reality Numbers– Precision
Fractional part - safely rounded to 36 digits (Hr) with no loss of verifiable meaning in reality
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Redefining the Reals
Completed Reality Numbers– Scale
Precision in reality is 36 decimal digits.
Precision for Completed Reality Numbers is 36 decimal digits.
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Redefining the Reals
Precision Output Strings– Definition
Includes all irrational numbers Generated by non-terminal processes Includes all ideas that generate
infinitely long sequences of digits
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Redefining the Reals
All Precision Output Strings become Reality Numbers
Precision Output Strings– Precision and Scale
Currently – an infinite number of decimal digits
ALL DECIMALS TO PRECISION OF 36 DECIMAL
PLACES
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Redefining the Reals
Reality Number Line– An updated Real Number Line– Precision – 36 decimal digits– Accuracy – No loss of verifiable
meaning in reality
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Christopher Tate
NGU Baseball Enumeration of
rational numbers Infinite Precision
Resolution Enumeration of
processes Results
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Enumerating the Rationals Example Cross-products
of denumerable sets are denumerable
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Eliminating Infinite Periodic Precision Periodic Reals have infinitely long
decimal expansions Example (1/7)10
– 0.142857142857142857142857… Eliminate the issue by changing the
base to the fraction’s denominator (1/7)10 = (0.1) 7 Radix is a presentation issue, not an
existence issue.
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Enumerating Text Strings Letters are
denumerable
As are words
As are sentences
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Enumerating Processes Processes
are denumerable.
One of these processes continually outputs Cantor’s Non-denumerable real.
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Enumerating Cantor’s Non-Denumerable Real
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Results
Periodic Rationals can be converted into reality numbers– By radix conversion.
Processes are denumerable. Cantor’s Non-denumerable Real
will be produced by one of the enumerated processes.
Reals are denumerable.
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Contessa Wright
Part 2: The Dismissal of
Cantor’s Diagonal Argument
Goodbye Contradictions
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The Need to Redefine the Real Numbers An infinitely long digit expansion
cannot be enumerated,– because it will never terminate.
There is no meaningful precision more precise than .
Yet classical real numbers include irrationals, with infinitely long digit expansions.
There is a need to redefine the real numbers.
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Cantor’s Failed Diagonal Argument The non-enumerated
real Is just a process
output Enumerated with the
process of ratio-of-integers p/q, where p ^ q are ε Z and q ≠ 0
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DeCantorizing the Argument There is no contradiction There is but one infinity There is no completed infinity 1.999... = 1 + .999…
1.999… = 1 + 9 * .111…1.999… = 1+ 9 * 1/91.999… = 1 + 11.999… = 2 X
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Anastasia Bridner
Honors Graduate Recipient of the
Excellence in Mathematics Award
Implications
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Implications
“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 ‘Babel-2’ of these sciences fall to pieces as a house of cards.” Alexander Zenkin
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Implications
Quantum Geometry Infinitesimal Polygons Resolving Paradoxes Redefining Functions Redefining Continuity Exact Integration
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Quantum Geometry
Hypotenuse of a right triangle with two sides of length 1 is not irrational.
It is a Reality Number rounded to a precision sufficient for a task.
Pythagorean Theorem produces approximations, not irrationals.
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Quantum Pythagorus
The hypotenuse of a right triangle with short sides of length 1 unit should be 1.41421… units,
which is not a choice when the units are infinitesimals
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Infinitesimal Squares
As the sides of a square approach the Planck limit,
The least square appears
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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
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Reality Math 2:1 Paradox Resolved There are
twice as many infinitesimals in a line segment of length 2 as there are in a line segment of length 1.
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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.
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Reality-Math Point Density Resolved Rounding b to the nearest
infinitesimalon the Reality Number Line shows that the relationship is many-to-one, not 1-to-1
ba
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
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Redefining Functions
A function must return a result Not a function :
– Y(X) = { 1, if x is rational -1, if x is irrational
}– Y( P ) will not terminate
A function of reality numbers, defined at a reality number, will always return a reality number.
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Redefining Continuity
Slopes |s| <=1 are continuous in quantum reality
A Continuity Delta spreads to the right
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Redefining Discontinuity
Slopes |s| >1 are discontinuous in quantum reality
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Integration is Exact
Integration is exact in quantum reality
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Discontinous Integration
Even Discontinuous Integration is exact in quantum reality
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Monica McAbee
Physics Facilitator Group Leader
Science Division Work Study
Summary Conclusions
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Summary
Periodic numbers– Can be transformed into Reality
Numbers Numbers with infinite expansions
– Are not reality numbers– But outputs from processes
Processes are denumerable Reality numbers are denumerable
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Conclusion
There is only one infinity – Not an infinity of infinities
Transfinite mathematics may be ignored
We have graduated into– The Quantum Mathematical
Universe
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A Final Thought…
Ptolomy once contended that the Universe is Earth-centered, but he was discredited.
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Now we notice that…
The Knowable Universe that expands at the speed of light…
Is Earth-Centered, …as Ptolomy
once contended Some things
never change
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The Beginning