Mathematics 4390

Senior SeminarOn Infinities

Dr. Brian L. Crissey

Chief Rabble Rouser

Playing the role of– Martin Luther

Co-conspirators

Andrew Blalock Anastasia Bridner Jessica Farmer Karla Kirby Monica McAbee Eric Moeller Delecta Rollins Christopher Tate Contessa Wright

Delecta Rollins

Who are we? What are we

tying to accomplish?

Carl Friedrich Gauss

“Mathematics is the queen of all sciences, and the Theory of Numbers is the queen of mathematics."

Georg Cantor

Cardinality Multiple

Infinities Insane

– Or a genius?

David Hilbert

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

Driven from Paradise

Reforming the Cantorian Church of PolyInfinitism

Andrew Blalock

Male Scholar Athlete of the Year (2006)

Philosophy Georg Cantor

New Ideas

How have they been received in the past?

Ptolomy’s Universe

Earth-centered

Stars fixed

Copernicus’s Universe

Heliocentric Kept silent Died before

reaction

Bruno’s Universe

Infinite No center

Outspoken Burned at the

stake

What Will Be Our Fate?

Ignored Scoffed at Castigated Accepted

Philosophies

Plato Platonism Monism One truth

Philosophies

Pluralism Many Truths No absolute

truth

Our Philosophy

Pragmatism Earth-

relevant What makes

sense On a finite

planet In a

Quantum Universe

Gottfried Wilhelm Leibnitz “Drawing is a very

useful tool against the uncertainty of words.”

So we will be as visual as possible.

Georg Cantor

The man The

mathematician

His contributions

His controversies

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.

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.

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.

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.

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

Cantor’s Diagonal Argument

Enumerating the reals

The non-enumerated real

The contradiction The conclusion Completed Infinities Multiple infinities

Jessica Farmer

Mathematics MajorSpanish Minor

Graduating in December 2007

Literature Review

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

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

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...”

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.

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.”

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

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.”

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.”

Eric Moeller

Winner of “Math Major w/ Best Style”

Loves Math as much as the 80’s

Reality Extrema

Reality Mathematics for a Finite Planet

The Religious War– Kronecker vs. Cantor

Reality Extrema The Planck limits Infinite Precision

Zeno’s Dichotomy

The infinitely large The infinitely small

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."

Time Limits Length

Planck Limits

Quantum-scale limits– Mass– Length– Area– Volume– Time

Smallest Meaningful Length New Jersey is to a Proton As a Proton is to a Planck length

Delta Infinitesimal

The X of integral calculus is the quantum limit of (X2 - X1)

is the legendary infinitesimal

X1

X2X

The Smallest Meaningful Length is the limit of

measurability. It is the limit of X in the differential quotient of Calculus.

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.

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…

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

Karla Kirby

“Class Chaplain” Mathematics

Major Spanish Minor Graduating

December 2007 Restructuring

the Reals The Reality

Number Line

Current Mathematics

One Real Number Line– Includes Rational Numbers– Includes Irrational Numbers

What is the problem?

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

The Problem

Where is the “…” on our current Real Number Line?

Redefining the Reals

Reality Numbers– Completed Reality Numbers– Process Output Strings

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

Redefining the Reals

Completed Reality Numbers– Precision

Fractional part - safely rounded to 36 digits (Hr) with no loss of verifiable meaning in reality

Redefining the Reals

Completed Reality Numbers– Scale

Precision in reality is 36 decimal digits.

Precision for Completed Reality Numbers is 36 decimal digits.

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

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

Redefining the Reals

Reality Number Line– An updated Real Number Line– Precision – 36 decimal digits– Accuracy – No loss of verifiable

meaning in reality

Christopher Tate

NGU Baseball Enumeration of

rational numbers Infinite Precision

Resolution Enumeration of

processes Results

Enumerating the Rationals Example Cross-products

of denumerable sets are denumerable

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.

Enumerating Text Strings Letters are

denumerable

As are words

As are sentences

Enumerating Processes Processes

are denumerable.

One of these processes continually outputs Cantor’s Non-denumerable real.

Enumerating Cantor’s Non-Denumerable Real

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.

Contessa Wright

Part 2: The Dismissal of

Cantor’s Diagonal Argument

Goodbye Contradictions

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.

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

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

Anastasia Bridner

Honors Graduate Recipient of the

Excellence in Mathematics Award

Implications

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

Implications

Quantum Geometry Infinitesimal Polygons Resolving Paradoxes Redefining Functions Redefining Continuity Exact Integration

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.

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

                 

                 

                 

                 

                 

                 

                 

                 

                 

Infinitesimal Squares

As the sides of a square approach the Planck limit,

The least square appears

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

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 Resolved There are

twice as many infinitesimals in a line segment of length 2 as there are in a line segment of length 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

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

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.

Redefining Continuity

Slopes |s| <=1 are continuous in quantum reality

A Continuity Delta spreads to the right

Redefining Discontinuity

Slopes |s| >1 are discontinuous in quantum reality

Integration is Exact

Integration is exact in quantum reality

Discontinous Integration

Even Discontinuous Integration is exact in quantum reality

Monica McAbee

Physics Facilitator Group Leader

Science Division Work Study

Summary Conclusions

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

Conclusion

There is only one infinity – Not an infinity of infinities

Transfinite mathematics may be ignored

We have graduated into– The Quantum Mathematical

Universe

A Final Thought…

Ptolomy once contended that the Universe is Earth-centered, but he was discredited.

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

The Beginning