Outline



The Anatomy of an Antinomy



Some properties and theorems of numbers relevant to topics
in the book THE EMPEROR'S NEW MIND by Roger Penrose

Glenn Johnston
(707) 664-0303
glennj@sonic.net

Part one




Different number sets.
Proofs and validity
Some math basics:
Primes
Countable numbers
Computable numbers
Transcendental numbers
Cardinal Numbers
Infinity
Transfinite numbers

Coding
Modeling
Mapping
Turing machine
Searles' Chinese room
Turing test


Part two



Cantor's Aleph-null and Aleph-1
Russell's Paradox
Richardson Antinomy
Gödel's Proof




Number Sets

N ={0, 1, 2, ...}
Z ={...,-3, -2, -1, 0, 1, 2, 3, ...}
Q ={ p/q| p, q,belongs to Z and q not equal to 0}
R ={ d0.d1d2... | d0 belongs to Z and the di are integers between 0 and 9, for i > 0}
C = {a + b*i | a,b belongs to R and i2 = -1}

Proof by Modeling

Find a model that visually demonstrates the consistency of the following postulates.
  1. Any two members of K are contained in just one member of L .
  2. No member of K is contained in more than two members of L.
  3. The members of K are not all contained in a single member of L.
  4. Any two members of L contain just one member of K
  5. No member of L contains more than two members of K


The following number can be considered a "mascot number" of the Concept Exchange Society, in that no person or group in the history of the world has ever looked at it or thought about it:
112358132134551873654266403038545987654324133
Why is this a valid statement?


In 1899, Giuseppe Peano axiomatized the arithmetic of cardinal numbers. His axioms can be stated as follows:
  1. Zero is a number.
  2. The IS (immediate successor) of a number is a number.
  3. Zero is not the IS of a number.
  4. No two numbers have the same IS.
  5. Any property belonging to zero, and also to the IS of every number that has the property, belongs to all numbers.


What follows is not an entirely faithful representation of Glenn's 'Outline' because of difficulties in rendering the mathematical symbols into HTML language. It will be improved with time. For example, dashes (-) in the text replace many of the esoteric mathematical symbols used by Glenn. Distortions are due to this fact, not to Glenn.

Marvin



GöDEL NUMBERS

Constant SignsGödel NumberMeaning
~ 1 not
v 2 or
_ 3 if ... then
_ 4 There exists a ...
= 5 equals
0 6 zero
s 7 The immediate successor of
( 8 punctuation mark
) 9 punctuation mark
, 10 punctuation mark


Numerical Gödel
Variable Number Possible Substitution Instance

x 11 0
y 13 s0
z 17 y

Numerical variables are associated with prime numbers greater than 10.
Sentential Gödel Possible Substitution Instance

Variable Number

p 11_ 0=0
q 13_ (_x) (x=sy)
r 17_ p_q
Sentential variables are associated with the squares of prime numbers greater than 10
Predicate Gödel Number A Possible
Variable Substitution Instance

P 11_ Prime
Q 13_ Composite
R 17_ Greater than
Predicate Variables are associated with the cubes of prime numbers greater than 10
Consider a formula of the system, e.g. '(_x) (x=sy)'. Literally translated, this reads: 'There is an x such that x is the immediate successor of y', and says, in effect, that every number has an immediate successor. The numbers associated with its ten constituent elementary signs are, respectively, 8, 4, 11, 9, 8, 11, 5, 7, 13, 9.

( _ x ) ( x = s y )
_ _ _ _ _ _ _ _ _ _
8 4 11 9 8 11 5 7 13 9

It is desirable, however, to assign a single number to the formula rather than a set of numbers. This can be done easily. We agree to associate with the formula the unique number that is the product of the first ten primes in order of magnitude, each prime being raised to a power equal to the Gödel number of the corresponding elementary sign. The above formula is accordingly associated with the number 2^8 * 3^4 * 5^11 * 7^9 * 11^8 * 13^11 * 17^5 * 19^7 * 23^13 * 29^9; Let us refer to this number as m . In a similar fashion, a unique number, the product of as many primes as there are signs (each prime being raised to a power equal to the Gödel number of the corresponding sign), can be assigned to every finite sequence of elementary signs and, in particular, to every formula.

( m = 1456664081617094091977899382886498187891470181481887898950349321995516094737500000000, and has 87 digits).

Most of the above was taken, sometimes verbatim, from the book Gödel's Proof by Ernest Nagel and James R Newman. Published by New York University Press, 1960.



Turing Machine


The simplified Turing Machine has the following characteristics:
1) The tape, 2) Internal state (I.S.) 3)The program 4) The output
The tape consists of an unlimited series of zeros and ones, e.g ... 0000010110101010000....and is called upon by the machine and read. The tape can be moved forward or backward; the marks can be overwritten or left as is.
The internal state (symbolized by the letters I.S.) consists of a discrete finite set of counting numbers; 0, 1, 2, ... The output, like the input, is a series of zeros and ones, overwritten on the input .
The program (which Penrose refers to as a specific Turing Machine) might look like this:
00 _ 00R
01 _ 71R
10 _ 151L
50 _ 01stop
... ...
The symbols on the left side of the arrow might be thought of as the "IF" part of an operation, and on the right is the "THEN" part. The left side is the "Look-up" side. The small numeral is the INTERNAL STATE (I.S.) numeral. The large numeral is the data (input) numeral.

On the right of the arrow, the small numeral on the left is the new INTERNAL STATE numeral. The Large numeral is the Output numeral. To the right of the output numeral is the TAPE MOVING instruction.

HOW IT WORKS
Notice that there are two parts to the input.
1) Read one unit on the tape (either a '1' or a '0'), called the INPUT
2) Take notice of the current INTERNAL STATE and find out what to do from the program, e.g. if you read a '1' and the I.S. is 7, look for a '71 _ ...' in the program)
There are three parts to the output:
  1. Change I.S to new I.S. numeral on the left (may be the same as old)
  2. OVERWRITE (if different) TAPE numeral to a 1 or 0
  3. Move TAPE to l. or r. as indicated or STOP if indicated
So, using two pieces of information, the I.S. and the current data bit, you perform three actions, change (if necessary) the I.S., change, if necessary the data bit, and either move the tape (left or right as indicated) or stop.

This simple machine adds one to a unary number.
00_ 00R, 01_11R, 10_01STOP, 11_11R

Try adding one to three. ...0000011100000...

Penrose gives the following description of a Turing Machine that effects Euclid's Algorithm:
00_ 00R, 01_11L, 10_ 21R , 11_ 11L, 20 _ 100R, 21_ 30R, 30_ 40R, 31_ 31R, 40_ 40R, 41_ 50R, 50_ 70L, 51_ 61L, 60_ 60L, 61_ 11L, 70_ 70L, 71_ 81L, 80_ 90L, 81_ 81L, 90_ 20R 91_ 11L, 100_ 00Stop, 101_ 101R

To find the highest common factor of 6 and 8, run this tape: ...000111111011111111000...

Important Numbers


666 - Number of the beast
668 - Neighbor of the beast
660 - Approximate number of the Beast
DCLXVI - Roman numeral of the Beast
666.0000 - Number of the High Precision Beast
0.666 - Number of the Millibeast
1/666 - Common Denominator of the Beast
666i - Imaginary number of the Beast
1010011010 - Binary of the Beast
1-666 - Area code of the Beast
00666 - Zip code of the Beast

from a web page -- Alan "Uncle Al" Schwartz > UncleAl0@ix.netcom.com ("zero" before @)

Bibliography

The Emperor's New Mind Roger Penrose Oxford University Press 1989 Gödel's Proof Earnest Nagle & James R. Newman New York University Press 1958 Gödel, Escher, Bach: Douglas R. Hofstadter Vintage\ Random House 1979, Mathematical Ideas Miller, Heeren, Hornsby Scott, Foresman, & Co. 1968
Journey Through Genius William Dunham John Wiley & Sons, Inc. 1990

Proof of Irrationality of the Square Root of Two

The proof relies on two very solid concepts , i.e. the fundamental theorem of arithmetic, which states that every integer has an unique factorization, and the simple fact that a fraction (ratio) may be considered to be in its lowest terms: the numerator and denominator have no common factor. Looking at it another way, any finite ratio A/B may have both terms repeatedly divided by any common factors a finite number of times until it has no common factors.


The proof starts with the assumption that some fraction, A/B, in its lowest terms, is the square root of two and proceeds to show that this is not possible. Without the fear of introducing anything like extraneous roots, we call the proposed square root of two, A/B

(1) __ = A/B Squaring both sides,
(2) 2 = A2/B2 , therefore,
(3) 2B2 = A2
Clearly, the left side is some composite integer with at least one factor of two. Since the exact same integer, on the right, is expressed as A2, A itself has to have a factor of two. Let A = 2K, where K is simply A/2. A2 = 4K2 Squaring and substituting in (3), we have
(4) 2B2 = 4K2 , or B2 =2A2. But now we have a similar situation where there is a factor of two on the right side, and therefore B must have a factor of two; but both A and B having the same original factor of two violates our premise that A/B was in its lowest form. Any substitution for B = 2L would lead to dividing both A and B by an infinite regression of twos. Therefore, no ratio A/B can ever be the square root of two.