ETT-R&D Publications E. Terrell
IT Professional, Author / Researcher August 2001
Internet Draft
Category: Proposed Standard
Document: draft-terrell-simple-proof-support-logic-analy-bin-02.txt
Expires February 13, 2002
The Simple Proof Supporting the Findings from the Logical
Analysis of the Binary System Which disposes the
Logical Dispute fostered by Modern Interpretation
for Counting in Binary Notation
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E Terrell [Page 1]
Logical Analysis of the Binary System February 13, 2002
TABLE OF CONTENTS
Abstract
Introduction: Beginning the Analysis
Chapter I: Broadening the Analysis of the Comparison
Chapter II. Setting the Stage by Defining the Parameters
Chapter III: Determining the Structure of Enumeration; The
Counting in the Binary System
Chapter IV: Comparing the Results from 2 Different
Numbering Systems
Chapter V: The Logician's Dialectic; Finalizing The
Comparison
Appendix I: 2 Binary Systems? True, False, or Enlightenment
References
E Terrell [Page 2]
Logical Analysis of the Binary System February 13, 2002
Abstract
The simple proof outlined in this presentation, regarding the
Logical Analysis of the Binary System, and the argument
concerning the Logical Derivation of the New Method for
Enumeration, which disputes the Modern Interpretation for
the Binary System, as being illogical and incorrect. Is in
fact, No proof at all. What I am presenting is a Comparative
Analysis; "Modern Interpretation" vs. "My Results", which
poses 2 questions: "Well, how do you begin your count? I
mean, if there are 5 objects to be counted, would your count
start with 'Zero' or 'One'?".
Furthermore, while the first version of this draft invoked a
negative response. That is, an acknowledgment was reached,
regarding the existence of another Binary System. However,
no one concurred with the results claiming the expulsion of the
Modern Interpretation. Nevertheless, this is an argument, and
the results derived there from, reserved for the venue of the
Appendix, which sets the tone to resolve such questions as:
"Do we now have 2 Binary Systems, establishing a slightly
different, and yet, equal relationship with the Set of
Integers? I mean, what do we have here? Is it possible
to have 2 distinct Binary Systems, whose difference
represents a different 'One-to-One Pairing' with the
Integers? Or are we to try once again, and decide, which
one of the two Numbering Systems actually represents a True
Binary System?"
Nevertheless, in either case, it should be stated that the
conclusions from this work produce an unquestionable outcome:
"The System of counting presently being used is a UNARY System,
from which the sequence of Counting begins with the Number '1',
and continues its progression using successive additions of the
Number '1' to derive the next or succeeding numbers. And while it
maybe called or labeled as being something different (i.e.
Decimal System), it is nevertheless Unary. Furthermore, while
Zero, '0', is used in every Numbering System, it is not itself,
a Number. It is only a symbolic notation, which represents
emptiness, or lack of an Object to which it refers. Hence,
Binary by definition, means '2', and nothing more. Therefore,
when considering the construction of any Numbering System that
employs or uses Binary Notation, we must first realize that the
first '4' numbers are derived from the Total Number of Possible
Unique Combinations, which are related to and derived from, the
Sequenced Numbers or Elements depicted as being Members of the
Binary Set. And further conclude, that all other succeeding
Binary Numbers are derived from these Combinations. In which
case, since the Binary Set equals {0,1}, the total number of
Unique Combinations equals the set {00, 01, 10, 11}."
E Terrell [Page 3]
Logical Analysis of the Binary System February 13, 2002
Introduction: Beginning the Analysis
To begin this task, we must first choose a beginning, and I
have chosen a Table, which represents a 'one-to-one'
relationship or pairing that expresses the Modern
interpretation of the relationship existing between the
Integers and the Binary Numbers. Where by, Table I
we have:
TABLE I
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only " 1's " and " 0's"
Depicting the Results from its current Presentation
Binary Representation Positive Integer
/ | \ / | \
1. 00000000 = 0 0
2. 00000001 = 01 1
3. 00000010 = 10 2
4. 00000011 = 11 3
5. 00000100 = 100 4
6. 00000101 = 101 5
7. 00000110 = 110 6
E Terrell [Page 4]
Logical Analysis of the Binary System February 13, 2002
TABLE I
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only " 1's " and " 0's"
Depicting the Results from its current Presentation
Binary Representation Positive Integer
/ | \ / | \
8. 00000111 = 0111 7
9. 00001000 = 1000 8
10. 00001001 = 1001 9
11. 00001010 = 1010 10
12. 00001011 = 1011 11
13. 00001100 = 1100 12
14. 00001101 = 1101 13
15. 00001110 = 1110 14
16. 00001111 = 1111 15
17. 00010000 = 10000 16
18. 00010001 = 10001 17
19. 00010010 = 10010 18
20. 00010011 = 10011 19
32. 00011111 = 11111 31
64. 00111111 = 111111 63
128. 01111111 = 1111111 127
256. 11111111 = 11111111 255
E Terrell [Page 5]
Logical Analysis of the Binary System February 13, 2002
Clearly, this can quite easily become a very long Table, and
an extremely time consuming endeavor regarding its
construction. But, if you are satisfied with its present
construction, and can fill in the missing entries using some
popular calculator, then I can proceed with the construction of
the Table yielding My Results. Given by, TABLE II, we have:
TABLE II
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Binary Representation Positive Integer
/ | \ / | \
1. 00000000 = 00 1
2. 00000001 = 01 2
3. 00000010 = 10 3
4. 00000011 = 11 4
5. 00000100 = 100 5
6. 00000101 = 101 6
7. 00000110 = 110 7
8. 00000111 = 111 8
9. 00001000 = 1000 9
10. 00001001 = 1001 10
E Terrell [Page 6]
Logical Analysis of the Binary System February 13, 2002
TABLE II
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Binary Representation Positive Integer
/ | \ / | \
11. 00001010 = 1010 11
12. 00001011 = 1011 12
13. 00001100 = 1100 13
14. 00001101 = 1101 14
15. 00001110 = 1110 15
16. 00001111 = 1111 16
17. 00010000 = 10000 17
18. 00010001 = 10001 18
19. 00010010 = 10010 19
20. 00010011 = 10011 20
32. 00011111 = 11111 32
64. 00111111 = 111111 64
128. 01111111 = 1111111 128
256. 11111111 = 11111111 256
E Terrell [Page 7]
Logical Analysis of the Binary System February 13, 2002
First, it should be clear from the basic rules of Set Theory,
that 'Any 0ne-to-One Pairing' between the objects representing
the Binary Set and those representing the Set of Integers, is
valid for both Tables. However, continued observation of the
Tables also reveals (using the comparison) a difference, which
exist between the Binary Representation and the associated
value represented by the Integers. In other words, the Binary
Representation is Paired or Associated with a different Value for
the Integer in each of the Tables. At this point, everyone would
wonder, which of the 2 Tables, pairing the Binary
Representation with one and only one Integer, is valid. And the
answer is; they both are valid. However, problem that must be
resolved, is a decision that bars arbitrary choice. That is, which
one of these 2 Tables can it be concluded, actually represents
an Equality existing between the Binary Representation and the
Integers, established by this 'One-to-One Pairing'?
Chapter I: Broadening the Analysis of the Comparison
We encountered a dilemma in the Introduction, because we
observed the creation of 2 distinct Sets, which are both, at this
point, equally valid representations for the Binary Method of
Enumeration (Counting). However, because of this difference,
we can only choose one of the methods depicted by the Tables,
which can be used when establishing a relationship of Equality
with the Integers. In other words, a One-to-One Pairing creating
Equality, can exist between any 2 Sets. Where the only
requirement is that, they each maintain the same Total Number
of Members. Nevertheless, using appropriate substitutions for
simplification, we can view this dilemma graphically. That is,
the Sets denoted by 'eq.1' and 'eq.2', can be mapped in a
'one-to-one' correspondence with the Integers, which is denoted
respectively by Figures 1 and 2.
Eq.1 {0,1,2,3,4,5,6,7,8,9}
Eq.2 {1,2,3,4,5,6,7,8,9,10}
E Terrell [Page 8]
Logical Analysis of the Binary System February 13, 2002
Fig 1.
1 2 3 4 5 6 7 8 9 10 = The Count of Total Number
-+-+-+-+-+-+-+-+-+-+ of Members in the Set
0 1 2 3 4 5 6 7 8 9 = The Elements or Members
Listed in the SET
Fig 2.
1 2 3 4 5 6 7 8 9 10 = The Count of Total Number
-+-+-+-+-+-+-+-+-+-+ of Members in the Set
1 2 3 4 5 6 7 8 9 10 = The Elements or Members
Listed in the SET
Now, comparing each of these Sets you will notice first, they
are not Identical, and second, when observing figures 1 and 2,
you will notice they are only Equal in the count representing
the total number Members they contain. But, only one of these
Sets can be paired in a 'One-to-One' correspondence with the
Set of Integers would produce an Identity in their definitions,
from which it can be concluded that an equality between the
members of both Sets exist. In other words, when counting the
Members of any Set, we can only associate the Members of
the Set with Number representing its placement within the Set,
or some Number representing the total Number of Members the
Set contains. This is only a comparison yielding some data
associated with Counting, which in this case, uses the Set of
Integers as the comparative tool for analysis. However, this
comparison does not produce nor establish an Identity by
definition, between the members of the Binary Set and the
members of the Set used in this comparison.
Nevertheless, before we can continue this analysis, we must
first agree to a Set of definitions, which will allow the necessary
distinction that will show the differences between these Sets.
Needless to say, without which, there can be No argument, nor
Distinction between the Sets we have created. Hence, the
evolution of 2 separate and distinct Methods of Enumeration for
the Binary System that would be Equally valid.
E Terrell [Page 9]
Logical Analysis of the Binary System February 13, 2002
Chapter II. Setting the Stage by Defining the Parameters
Can we all agree that Language is Ambiguous? And further
note that, without the Rules governing some mandatory
structure and definitions ascribing some meaning, Language
the Tool that it is, would have no viable purpose. In other words,
we could avoid the Trap that would ultimately befall us, if we
would adhere to, and follow the following definitions.
Definitions
1. A 'SET': A Set is a Collection of Objects, which list the
Members that can be arbitrarily chosen or included in the Set.
2. 'Binary Set': A Binary Set contains 2 Objects, which means
have or consisting of 2 things.
3. 'Integer': Any numerical representation, which is not a Fraction
and does not contain a Fractional Component in its description
or representation.
4. 'One-to-One' Mapping or Correspondence: Any Pairing Such
that, One and Only One Object is Compared with, or Mapped
to One and Only One Object, in such a way that this Pairing
Yields a Unique Object Consisting of Parts.
5. 'Zero': Any Object that has No Members, which has No Value,
and does not consist of any Parts.
6. 'Null Set': Any Set that is Empty, and contains No Members
or Elements.
7. 'Equality': A Relationship, which provide a means to establish
an Identity between 2 or more Objects being compared.
8. 'Definition': A Grouping of Words, Objects, Numbers or some
Combination derived there from, which ascribes some unique
Meaning to the Object in which the Definition refers.
E Terrell [Page 10]
Logical Analysis of the Binary System February 13, 2002
If we can now all agree to the Definitions given above, and
accept the meaning ascribe to the Objects they refer, then
we can move on to Chapter III.
Chapter III: Determining the Structure of Enumeration;
The Counting in the Binary System
Please be patient with me, because throughout this
comparison we will be reflecting back and forth between the
Chapters that we have already read. However, this is
necessary, because we cannot display, at least not on paper,
all of the data that we have accumulated, which results from
our agreements and the beginnings of our analysis.
Nevertheless, there are 2 things, which must immediately be
brought to the forefront of our discussion. First, observe that
figure 2 maintains a relationship, which by definition is Equality.
And this Equality represents the Identity, which exist between
the Members of the Set denoted by 'eq.2' and the Set of
Integers, or Counting Numbers (Whole Numbers).
Second, you should note, that while there exist a 'One-to-One
Correspondence' between the members of the Set denoted by
'eq.1' and the Set of Integers, there is No Equality. And this is a
valid observation when figure 1 is viewed, because by definition,
'Zero' cannot be assigned any Value. But, the question to be
imposed now is: 'How are these observations related to Tables
1 and 2?'. While, further analysis would yield the question:
'Could it be that, upon analysis, Table 1 would mimic 'eq.1', and
Table 2 would mimic 'eq.2'?'. Nevertheless, even if the answer to
the latter were true, it is not enough information upon which we
could assert that the Modern Method of Enumeration for the
Binary System is wrong! We must garner more facts to reach
this conclusion, because from the information we have
compiled thus far, clearly nothing is conclusive and neither
method represented in Tables 1 and 2, can be either
Right or Wrong.
E Terrell [Page 11]
Logical Analysis of the Binary System February 13, 2002
In other words, if we now pursue an analysis beyond the
'One-to-One Correspondence' we have used thus far, and
Equate, using the Results from Tables 1 and 2, the Members
from the Binary Set with Members from the Set of Integers, as
would be the results presented in Tables 1 and 2. Then what
we would achieve, would be a situation existing between the
Members of the Binary Set itself. That is, while the members
from the Set of Integers can be shown to be equal to each other
in a 'One-to-One Correspondence'. No such relationship can
exist between the Members of the Binary Sets displayed by
the 2 Tables in our analysis. This is because, the Members of
the Binary Set in each of the Tables, independently from each
other, has been assigned a Corresponding Value, creating a
unique mapping with the members from the Set of Integers,
which establishes a conflict between the values of the Members
from the Binary Sets in the respective Tables. This conflict
exist because, there is a difference between the assigned
value for the Integers associated with the members from the
Binary Sets, in each of these Tables.
But, how can such a dilemma be resolved? One way is by
actual testing, through the use of equations, a move beyond
the empirical analysis we have been using thus far. That is,
using the results from each of the Tables, we can create
equations, which can be used to validate our beliefs, one
way or the other.
Chapter IV: Comparing the Results from 2 Different
Numbering Systems
Before we can use any Equation, or different Numbering
System, we must first define them, and agree to the
Definition provided.
E Terrell [Page 12]
Logical Analysis of the Binary System February 13, 2002
Definition II
1. 'Binary Number': A 'Binary Number' is any Number, which
can be derived, using only the Members or Elements from
the 'Binary Set', {0,1}. A Binary Number is said to be a
Number of the Base 2.
2. 'Natural Number': Is a 'Whole or Counting Number' derived from
the 'Set of Integers', {0,1,2,3,4,5,6,7,8,9}. A Counting Number
is said to be a 'Dec' Number or Decimal Number of the base
10.
3. 'Equation': An 'Equation' is a Statement of Equality, which may
contain One or more Arithmetic Operators, that represents the
relationship between 2 or more Quantities.
Needless to say, if we agree upon the definitions from
Definition II, we can now create Equations, which would be an
equal representation of both the Binary Set and the Set of
Integers. This would produce the Results, as given by Tables
IIa and Ia, which are generated respectively, from Table II
and I. To it put more precisely, since the focus of our argument
is only the Binary Representation; Note the ongoing scenario.
That is, given the equations:
eq.3 X^B = I, where B = Exponent and I = result
or
eq.4 X^I = B, where I = Exponent and B = result
Note: Where "B = Binary representation", "I = Integer",
"F = Fraction" And "X" represents any Variable.
E Terrell [Page 13]
Logical Analysis of the Binary System February 13, 2002
In other words: "Is the Value of the Binary Representation
given by the Value of the Exponent or the Result of the
Equation itself?" Then, from Tables I and II we can generate
their respective Tables, Ia and IIa, which would provide us with
additional information. Where by:
TABLE Ia
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 0 0
2. 2^0 = 1 00000000 = 0 0
3. 2^1 = 2 00000001 = 01 1
4. 2^B = 2 00000010 = 10 2
5. 2^B = 3 00000011 = 11 3
6. 2^2 = 4 00000100 = 0100 4
7. 2^B = 5 00000101 = 0101 5
8. 2^B = 6 00000110 = 0110 6
9. 2^B = 7 00000111 = 0111 7
10. 2^3 = 8 00001000 = 1000 8
11. 2^B = 9 00001001 = 1001 9
12. 2^B = 10 00001010 = 1010 10
13. 2^B = 11 00001011 = 1011 11
14. 2^B = 12 00001100 = 1100 12
15. 2^B = 13 00001100 = 1101 13
16. 2^B = 14 00001110 = 1110 14
E Terrell [Page 14]
Logical Analysis of the Binary System February 13, 2002
TABLE Ia
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
17. 2^B = 15 00001111 = 1111 15
18. 2^4 = 16 00010000 = 100000 16
34. 2^5 = 32 00100000 = 100000 32
66. 2^6 = 64 00100001 = 100001 64
130. 2^7 = 128 01000000 = 1000000 128
258. 2^8 = 256 100000000 = 100000000 256
Just think! We have not completed the Table conversion for
Table II, and already you can see a problem, which can quite
easily be the conclusion that marks the End of the Modern
Presentation for the Method of Enumeration for the Binary
System. That is, the fall of the Modern presentation of the
method for enumeration in the Binary System, can
prematurely be concluded when viewing equations 1 and 2
from Table Ia, because there exist a Mathematical
Impossibility. However, prior to the hastily acceptance of any
such evidence as being the final straw, we must first agree
that the Binary and Integer Columns from Table Ia are the
same, which are Equal to the Binary and Integer Columns
from Table I.
Nevertheless, continuing the analysis, observe the information
corresponding to the Note under 'eq.4', which has 'X' defined as
a Variable; from Table Ia, equations 1 and 2, we have:
eq.5 X^0 = 0^0 = 0
E Terrell [Page 15]
Logical Analysis of the Binary System February 13, 2002
But! This is not possible, because equation 2 was defined as
being Equal to '1'. That is, given by 'eq.6', we have:
eq.6 X^0 = 1; Where 'X' is not Equal to '0'.
Furthermore, since the value of the Exponent in the equation
itself, can never produce a Result of Zero, and since any value
other than 'Zero' assigned to the Base having an Exponential
value equal to zero will always produce a Result Equal to '1'.
We can conclude that there has not been any Mathematical
Laws violated in this comparison, when 'X' was chosen as a
Variable. Furthermore, taking note of the fact equations 3
and 4 from 'Table Ia' are Equal, and that it has long since
been held by the definition of a 'Binary Number', that is, it is
DEFINED by an EXPONENTIAL Equation having a of Base 2.
Then we can conclude that the series of Binary Numbers
generated by Table I, and represented in Table Ia, is in fact
wrong, because it is not Logically consistent, nor does it
map in a 'One-to-One Correspondence' with the Equations
they were derived from.
In other words, you cannot derive the Binary Numbers listed
under the Binary Representation Column, from the Equations
listed under the Exponential Enumeration Column, which are
given in Tables I and Ia. Nevertheless, our comparison is not
finished, because there still remains Table IIa. Given by:
E Terrell [Page 16]
Logical Analysis of the Binary System February 13, 2002
TABLE IIa
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 0 0
2. 2^0 = 1 00000000 = 00 1
3. 2^1 = 2 00000001 = 01 2
4. 2^F = 3 00000010 = 10 3
5. 2^2 = 4 00000011 = 11 4
6. 2^F = 5 00000100 = 100 5
7. 2^F = 6 00000101 = 101 6
E Terrell [Page 17]
Logical Analysis of the Binary System February 13, 2002
TABLE IIa
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
8. 2^F = 7 00000110 = 0110 7
9. 2^3 = 8 00000111 = 0111 8
10. 2^F = 9 00001000 = 1000 9
11. 2^F = 10 00001001 = 1001 10
12. 2^F = 11 00001010 = 1010 11
13. 2^F = 12 00001011 = 1011 12
14. 2^F = 13 00001100 = 1100 13
15. 2^F = 14 00001101 = 1101 14
17. 2^4 = 16 00001111 = 1111 16
33. 2^5 = 32 00011111 = 11111 32
65. 2^6 = 64 00111111 = 111111 64
129. 2^7 = 128 01111111 = 1111111 128
257. 2^8 = 256 11111111 = 11111111 256
E Terrell [Page 18]
Logical Analysis of the Binary System February 13, 2002
Nevertheless, the analysis of 'Table IIa' produces the
necessary results, from which it can be concluded without
any doubts. That it expresses a relationship that represents
the 'One-to-One Correspondence' existing between the Results
from the Exponential Equation under Exponential Enumeration
Column, with the 'One-to-One Correspondence' existing
between the series generated and listed under the
corresponding and respective Binary Representation and
Positive Integer Columns. However, even this is not the
announcement of the Finality, because we have not ascribed
any Definitions, establishing some Equality. And there is still
the question of the assignment of the starting point, which
leads to the resounding question. 'What is the location and
definition of 'Zero' relative to the established relationship
generated by this 'One-to-One Correspondence'?'
In other words, when taking account the information derived
from the analysis and comparison of all 4 Tables, and the
foregoing conclusion regarding the choice of some starting
point related to 'Zero'. We could create 2 additional Tables,
derived respectively from Tables 'Ia and IIa', and eliminate the
Binary Representation Column and its respective members.
That is, we can change the focus of our analysis from the
Binary System, and construct 2 Tables as described above,
to determine the correct starting point for enumerating in the
Binary System when the Binary Representation Column and
its respective members were again included in our analysis.
Where by, given the respective Tables denoted by, Ib and IIb,
we have:
E Terrell [Page 19]
Logical Analysis of the Binary System February 13, 2002
TABLE Ib
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Enumeration Positive Integer
/ | \ / | \
1. 0^0 = 0 0
2. 2^0 = 1 0
3. 2^1 = 2 1
4. 2^1 = 2 2
5. 2^F = 3 3
6. 2^2 = 4 4
7. 2^F = 5 5
8. 2^F = 6 6
E Terrell [Page 20]
Logical Analysis of the Binary System February 13, 2002
TABLE IIb
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Exponential Enumeration Positive Integer
/ | \ / | \
1. 0^0 = 0 0
2. 2^0 = 1 1
3. 2^1 = 2 2
4. 2^F = 3 3
5. 2^2 = 4 4
6. 2^F = 5 5
7. 2^F = 6 6
8. 2^F = 7 7
Surely, anyone would notice the ambiguity, when looking for
any relationship existing between the members of the
respective Exponential Enumeration, and Positive Integer
columns. In fact, the analysis of Table 'Ib', noting specifically
rows 1 and 2, emphasizes the Mathematical Anomaly, which
prevents the existence by definition, of any relationship
denoting equality. Furthermore, it should also be very clear,
that there cannot exist a 'One-to-One Correspondence'
between the members of the respective columns either,
because there is no consistency nor continuity between the
individual entities comprising the rows listed under the
columns. In other words, while we can arbitrarily establish
equality between the Binary Numbers and the Integers, as
depicted in Tables I, Ia, and Ib, which would maintain a
'One-to-One' relationship with each other. The only equation
defined by the Laws of Mathematics, which would logically
support the existence of such a relationship, would be Linear.
Or the Identity Equation, which bars the existence of the
Binary Set. Hence, from the analysis of Tables I, Ia, and Ib,
which includes the Mathematical Laws and the Principles of
Logic, we cannot deduce nor derive any reason for the
existence of the Binary Method for Enumeration as presented
by its Modern interpretation.
E Terrell [Page 21]
Logical Analysis of the Binary System February 13, 2002
[e.g. The Identity Equation;
0 + 0 = 0,
1 + 0 = 1,
2 + 0 = 2,
... ,
255 + 0 = 255]
However, this is not the case for Table IIb, which clearly
displays a relationship existing between the members under
the respective Exponential Enumeration, and Positive Integer
columns. In other words, the Results generated by the
exponential equations under the Exponential Enumeration
column, map exactly in a 'One-to-One Correspondence' with
the Integers under the Positive Integer column. In fact, any
analysis of Tables II, IIa, and IIb, and the conclusions derived
above, would show this to be an unquestionable conclusion.
Nevertheless, if you now question the Assignment of Element
from the Binary Set, denoted by {0}, and represented as '00',
which was equated to '1'. Then I would call your attention to a
fact, which states that; 'Something cannot be derived from
Nothing!'. In other words, by definition, a Binary Set, is a Set
that contains 2 things, Members, or Elements. And while the
graphical depiction for this particular Member has the same
appearance as 'Zero' or the 'Null Set', it is not equal to either
of them: 'Nor can it be!'. Especially since by the definition,
'Zero' and the 'Null Set', cannot be used to create a Value
greater than itself. Hence, the Binary Element denoted by {0},
is not equal to either 'Zero' or the 'Null Set'. In other words,
there is No such thing as a Binary Set having only 1 Member,
as is the case depicted by Tables I, Ia, and Ib, because then it
would not represent a Binary Set. Furthermore, since the only
demand imposed by the Definition of the Binary Set, is that, it
must contain 2 Things. Then what they are, as long as they are
not empty, is a matter of arbitrary Choice. Therefore, the
Mathematically Correct and Logical Derivation, which is the
representation for the Binary Notation, that expresses the
process of Enumeration (Counting), is given by Table IIa.
E Terrell [Page 22]
Logical Analysis of the Binary System February 13, 2002
However, if you remain in doubt regarding the conclusion
above, please take note of the ongoing argument. Where by, it
should be understood, that any Numbering System can be
Derived, or Created, by First establishing a 'One-to-One
correspondence' of the New Numbering System with that of
the Integers (Or Any of the Set of Numbering Systems,
which exist as Member of the Real Number Set.), then by
setting each Mapped Entity Equal to the Corresponding Integer
to which it initially maintained the 'One-to-One Correspondence'
with. Now, we have created a New Numbering System. Next,
we must TEST our New System using the Same Laws Derived
from Mathematics, which are valid with the Numbering System
we used to create our New Numbering System. If our New
Numbering System Fails Under any of the Mathematical
Laws, which are valid for the Numbering System that was used
to create our New Numbering System, then we can conclude
that the New Numbering System we Created is Invalid,
Illogical, and Wrong.
Chapter V: The Logician's Dialectic; Finalizing The Comparison
First and foremost, we must agree to the present method as
being the definition for the elements of the Binary Set; {0,1}.
Please note the equation, as given by "eq.7 ":
eq.7: Binary 0 = the Integer
This is an established fact, as given in every explanation
concerning Binary Enumeration. Which clearly means that the
{0} element of the Binary Set, {0,1}, is equal to the Integer 0.
The problem concerning this definition is that: "If any element
or member of the Binary Set, defined by {0,1}, is equal to the
Integer 0. What happens to the definition of the Null or Empty
Set? "? In other words, if the Integer 0 implies the existence
of a Nothing State, or the Condition of having Nothing, then
there is a Contradiction between the Definitions of Existence
and Non-Existence; having and Not having.
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Logical Analysis of the Binary System February 13, 2002
Clearly this defines the perception of Dr. Warren Heisenberg
and his Uncertainty Principle of Classical Physics. When trying
to measure the Position and or Velocity of an Electron without
effecting or changing either of these parameters during the
analysis. Unfortunately, this particular situation is not as
complex, because the tools and the object under consideration
have no motion in which we would disturb, thus destroying our
analysis. However, we must decide upon the correct definitions
existing in Set Theory and those of the Set of Integers, which
would resolve the dilemma regarding the definition of the Binary
element denoted by {0}.
In which case, the Zero Integer must be equated to the Null or
Empty Set. Especially since, they both imply by definition,
having Nothing, or the State of Emptiness. Consider for a
moment, if you will, the foregoing example:
"Let the Null Set {0} equal the Integer 0; as given by 'eq.8':
eq.8 {0} NULL or Empty Set = 0"
[From which it can be easily established that
Binary 0 = NULL SET. Especially since, it
follows by conclusion from the Axioms for Equality
that if "A + B = Z" and "B + C = Z", then
"A + C = Z": and "A = C". Hence, "eq.7 = eq.8".]
Now, by definition, if this is indeed true, then "eq.8" is empty
and has absolutely No Members. In which case, " eq.7 is
equal to eq.8". Right? However, if this were the case, then the
Binary Set would, and should be equal to the set "{1}", which
has only One Member. Because enumerating the elements of
any set means counting the Total Number of Members
contained in the Set itself. And since, the Null element
(Null Set) can be considered a member of every Set. If it is
indeed empty, it is not countable, because there is nothing
to count. It is, by definition, representing Absolutely nothing.
In other words, you cannot count Nothing as being Something!
That is, given by 'eq.9', the total Number of members or
elements in the Binary Set must be equal to 2, because the
definition of Binary, means 2.
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Logical Analysis of the Binary System February 13, 2002
eq.9 {0,1} = 2
But, since the results of "eq.7 and eq.8", are those currently
maintained and derived from the established definitions of Set
Theory and those concerning the Binary System. Zero in its
universal Definition, is indeed the Null Set, which means or
implies a State of having nothing or No Elements. This would
then imply that the actual representation of 'eq.9' would be
that depicted by 'eq.10':
eq.10 {0,1} = 1 = {1}
Now, we come full circle, because if "eq.9 is False" and
"eq.10 is True", or if "eq.9 is True" and "eq.10 is False". Then,
not only we have lost the Definition of Binary, which is not
equal to, nor does it mean Unary. But, we also loose the
Definition of Zero. This is because if you give a Value to
Zero, then it is no longer Nothing, Void, or Empty. It would truly
be or become Something, and that Something would be other
than that which does the Definition of Zero itself define.
Notwithstanding however, following the Current Definitions, or
that which has been established for the Binary System. If you
believe that which was established and given by "eq.9 ", as
being correct, then that which can be deduced or concluded
from the laws governing Set Theory, given by "eq.11", would
also be true. Where by;
eq.9 {0,1} = 2
(Where 2 is the total number of members of the SET)
Then
eq.11 {0} = 1 and {1} = 1, because {0} u {1} = {0,1} = 2
(Where " U " represents the "UNION of the Sets... And the
" 1 " on the Right Side of the Equal Sign represents the
Total Number of Members [Count] Contained in the Set.)
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Logical Analysis of the Binary System February 13, 2002
Which is indeed a contradiction! Because "eq.7", as well as
"eq.8" maintains that "{0} = 0". And because the "Union" of
any Set with the Null Set represents the "fundamental principle
of Identity", the Null Set cannot be counted. Therefore, Binary
Zero is 'Not Equal' to the Integer Zero, 0, nor is it Equal to the
Null Set! And 'eq.11', as well as Table IIa, represents the true
and accurate depiction of the Binary Set, denoted by {0,1}.
Now, oppose the presentation I have argued thus far, which
provided a valid and logical reason, rendering the necessary
justification for the rewriting the Method of Enumeration for the
Binary System. And allow Binary Zero to Equal the Integer Zero!
That is, assume that in all cases, 'e.g. 7, 8, and 9' are
unconditionally True. Then this argument would now focus upon
deciding, according to 'Tables Ia and IIa', which Value the
Binary Representation actually represents. That is, given the
equations:
eq.3 X^B = I = R,
where B = Exponent and I = result
or
eq.4 X^I = B = R,
where I = Exponent and B = result
Note: Where "B = Binary representation",
"I = Integer", "F = Fraction" and
"X" represents any Variable. And in this case,
"R" and "X" can never be equal to "F"
(Some Fraction).
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The problem here however, when reviewing each of the Tables,
since the Binary Number has been mapped to represent some
Integer, is choosing the Table, which accurately represent the
results depicted in the Column of Exponential Equations. That
is, when results are compared with the results of the Columns
representing the Binary Number mapped to some Integer. You
will note however, that each Table uses a different equation to
start or initialize their respective mappings. That is, Tables II
and IIa consistency uses 'eq.3', which is the result of the
equation representing an Integer. While Tables I and Ia, uses
'eq.4'. This clearly causes a problem! Especially since, Tables
I and Ia use both 'e.g. 3 and 4', eventually, to represent the
relationship between the Binary System and the Integers.
Which clearly shows no direct mapping or count with the
total number of members in the Binary Set and the Set of
Integers Represented by the Number Line. And this fact is
established by equations 1 and 2 under the Exponential
Enumeration Column of Table Ia.
Where by, in both cases these equations center upon the
value of the Exponent only. That is, not until equation 4,
which changes the emphasis to that of the Result, that is
governed by 'eq.3'. But, the problem with this method is that,
equations 4 and 5 under the Exponential Enumeration Column,
does not represent a Binary or Integer format, which was
derived from using either 'e.g. 3 or 4'. If it did, it would be a
repeat of equations 6 and 10 under the Exponential
Enumeration Column. Now wouldn't it?
I mean, what does the Binary Number Represent? Is it the
Exponent in the equation? Or! Is it the Result? Clearly, this
shows that the count or consistency between the Binary
System and the Set of Integers has lost its logical Continuity,
or that somebody has Just Plain Committed an Error.
Nevertheless, these conditions do not exist in the results
given by Tables II and IIa, because it consistently uses 'e.g. 3',
which consistently maps the Result from the Exponential
Enumeration Column with the respective Mappings of the
'One-to-One' Correspondence existing between the
Binary Numbers and the Integers.
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Hence, we can now conclude, Zero is a Set, which is a
Subset of Every Set, and it is a Universal Set itself, that
cannot have any value ascribed, because it has no value
at all. What this means is that, Zero or the Null Set is a
Subset of every Set and a Proper Subset of every Set except
itself. In other words, while you can use Zero to represent the
Null Set, and include it as a member of any Set. Only when it
is a noted and visible Member, can it be counted. However, this
count cannot ascribe any Value to Zero or the Null Set beyond
the one-to-one Total, which is a count representing the Total
Number of Visible Members Contained in the Set itself. In
other words, Tables II and IIa represent the True and accurate
depiction of the Binary Numbers, which are paired in a
'One-to-One Correspondence' in a relationship denoting
Equality, with the Integers. Therefore, the Modern
representation of the Binary Numbers, and its Method for
Enumeration is indeed, unquestionably wrong.
Appendix I: 2 Binary Systems? True, False, or Enlightenment
It would not be, nor could it ever become the End, if the
light at the end of the tunnel was to dim or go out. I
mean, you would continue your trek, and assume that
only the night has caused the darkness, because the
Sun has set. Wouldn't you? With this in mind, let's
untangle, and delve deeper into the mysteries, now
plaguing the Binary System.
Beginning our quest however, accept as being only one
side of the truth, the conclusions associated with the
results presented by Tables II, IIa, and IIb. And accept
as being the rigor establishing only the foundation for the
argument in opposition, a partial truth, which is
represented by the conclusions associated with the
information derived and established by Tables I, Ia, and
Ib. In other words, further analysis would not only result
in another Table depicting a different view of the Modern
Interpretation of Binary Enumeration, previously
represented by Tables I, Ia, and Ib. But, it would also
enhance and strengthen the acceptance of the foundation
derived for the Alternate View of the Binary System.
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Logical Analysis of the Binary System February 13, 2002
However, prior to any forthright Construction of Table Ic,
following in sequence from Tables I, Ia, and Ib. It
would facilitate the analysis of the logical argument, if we
first reiterate the requirements that were logically developed,
that established the foundational definitions and
requirements, which would be the mandate for any Binary
System to exist.
Binary Principles
1. Binary; Consisting of 2 Things, Elements, or Members.
2. Zero and the Null Set are implied by the same definition
3. Zero; Having no Quantity, Size, Members, or elements;
representing a State of Condition of Nothingness.
4. Binary Set; Consisting of 2 and only 2, Elements or
Members.
5. Union of Set; Combining the Elements or Members of 2 or
more Sets, resulting in 1 Set containing the total, which
represents the combined total of the Members from the
initial Sets.
6. 'Equality': A Relationship, which provides a means to
establish an Identity between 2 or more Objects being
compared.
7. Binary Zero is represented by '00', since it is not empty, it
is not equal to either the Zero Integer or the Null Set.
Now if you are satisfied with the list of Principles derived from,
and associated with the Binary System, with the exception of 7.
We can construct Table Ic, which represents another view for
the Modern Method of Binary Enumeration.
E Terrell [Page 29]
Logical Analysis of the Binary System February 13, 2002
TABLE Ic
"The Modern Interpretation of the Binary System of
Enumeration" Counting, using only "1's" and "0's"
Depicting the Results from its current Presentation
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 00000000 = 0 0
2. 2^0 = 1 00000000 = 01 1
3. 2^1 = 2 00000001 = 10 2
4. 2^F = 3 00000010 = 11 3
5. 2^2 = 4 00000011 = 100 4
6. 2^F = 5 00000100 = 101 5
7. 2^F = 6 00000101 = 110 6
Notice that Table Ic maintains the 'One-to-One' validity as Table
IIa. However, as with Tables I and II, their differences remain the
same. In fact, any comparison with Table IIa maintains the
same validity, except for their different starting points. In other
words, Table Ic and Table IIa are 2 distinct Numbering Systems,
that use the Binary Notation in a 'One-to-One Pairing' with the
Integers to define and establish equality.
"Do we now have 2 Binary Systems, establishing a slightly
different, and yet, equal relationship with the Set of Integers?
I mean, what do we have here? Is it possible to have 2
distinct Binary Systems, whose difference represents a different
'One-to-One Pairing' with the Integers? Or are we to try once
again, and decide, which one of the two Numbering Systems
actually represents a True Binary System?"
E Terrell [Page 30]
Logical Analysis of the Binary System February 13, 2002
TABLE IIa
"The Reality of the Binary System of Enumeration"
And the Series Generated when Counting, using
only " 1's " and " 0's "
Exponential Binary Positive
Enumeration Representation Integer
/ | \ / | \ / | \
1. 0^0 = 0 0 0
2. 2^0 = 1 00000000 = 00 1
3. 2^1 = 2 00000001 = 01 2
4. 2^F = 3 00000010 = 10 3
5. 2^2 = 4 00000011 = 11 4
6. 2^F = 5 00000100 = 100 5
7. 2^F = 6 00000101 = 101 6
Following the same investigative analysis used in earlier
chapters, we can depict this difference graphically. That is,
if we were now to extrapolate from the respective Binary
Notations, as it would be given by the Integers' additive
method of progression, which produces the counting series
using successive additions of 1. We could then generate a
number line, denoting a 'One-to-One Mapping' with the
Integers that would more accurately depict these noted
distinctions. Given respectively by figures 3 and 4, we have:
Fig 3.
1 2 3 4 = The Count of Total Number
-+-+-+-+ of Members in the Set
0 1 2 3 = The Elements or Members
Listed in Table Ic's Binary Set
E Terrell [Page 31]
Logical Analysis of the Binary System February 13, 2002
Fig 4.
1 2 3 4 = The Count of Total Number
-+-+-+-+- of Members in the Set
1 2 3 4 = The Elements or Members
Listed in Table IIa's Binary Set
What the bottom row of numbers actually represents, is the
total number of combinations, which will be generated from the
Binary Set, {0,1}. However, these combinations are used in a
way similar to the way the '1' is used in the Integers, which
increments from right to left using and changing only the ' 0 or
1' notations from the Binary Set to generate a series of Binary
Numbers. In other words, they generate a series governed by
the operation of addition. That is, given respectively by figures
5 and 6, we have:
Fig 5.
{01}, {10}, {11}
2 3 4
Fig 6.
{00}, {01}, {10}, {11}
1 2 3 4
Well, how do you begin your count? I mean, if there are 5
objects to be counted, would your count start with 'Zero' or
'One'? Clearly, the Set of Integers from which the Counting
Numbers were derived, was only a graphical depiction, to be
used in such a way, as to render a picture of the Number to
be represented, which used one or more of these members
to achieve the desired result. And nothing more. In other words,
the Set of Integers or Whole Numbers, maintains the additional
distinction of being a short-hand representation for the Operation
of Addition, from which the sequence of Numbers is derived from
the Unary Set {1}.
E Terrell [Page 32]
Logical Analysis of the Binary System February 13, 2002
Furthermore, I am sure you observed from figure 5, that the
equating of Binary Zero to the Integer Zero reduced the number
of combinations resulting from the Binary Set. Which is
actually the cause which produces the SHIFT in the
'One-to-One Pairing' with the Integers. I mean, the assignment
of the Beginning Point for any Numbering Systems is very
important, because it sets the starting point that will be used
for counting.
Moreover, further analysis of the resulting Combinations
derived from both of the respective Binary Sets, using Tables
Ic and IIa. Clearly shows the equality existing between each of
these Sets, which is derived from the 'One-to-One Pairing'
equating the Points on the Number Line, denoting the
Integers, with the Binary Notations they respectively represent.
If however, we mapped the results indicated by figures 5 and 6,
using the respective mappings given by figures 3 and 4, we
would establish the necessary proof for concluding, that the
method derived for Counting using the Modern Interpretation is
wrong. In other words, any 'One-to-One Mapping' with the
Integers and the Combinations resulting from figures 5 and 6,
would clearly show that the missing Set, given by the
Combination {00}, would result in a inaccurate mapping
denoting an Inequality with the Sequence of Counting Numbers
derived from the Set of Integers; that is, the Set of Counting
Numbers denoted by: {1,2,3,4,5,6,7,8,9,10}. In which case,
the Universal Set " I ", for the Integers, would equal the Set
denoted by:
Fig 7.
x|x is an element of I = Integers
{ {...-10,...-5,-4,-3,-2,-1} {0} {1,2,3,4,5,...,10} }
Where its number line mapping is given by:
Fig 8.
-10 + -9 ... -5 +... -2 + -1 + 0 + 1 + 2 + 3 ... 5 +... + 10
E Terrell [Page 33]
Logical Analysis of the Binary System February 13, 2002
Nevertheless, the System of counting presently being used is a
UNARY System, from which the sequence of Counting begins with the
Number '1', and continues its progression using successive
additions of the Number '1' to derive the next or succeeding
numbers. And while it maybe called or labeled as being something
different (i.e. Decimal System), it is nevertheless Unary.
Furthermore, while Zero, '0', is used in every Numbering System
(denoting its' universal application), it is not itself, a Number.
It is only a symbolic notation, which represents emptiness, or lack
of an Object to which it refers. Hence, Binary by definition, means
'2', and nothing more. Therefore, when considering the construction
of any Numbering System that employs or uses Binary Notation, we
must first realize that the first '4' numbers are derived from the
Total Number of Possible Unique Combinations, which are related to
and derived from, the Sequenced Numbers or Elements depicted as
being Members of the Binary Set. And further conclude, that all
other succeeding Binary Numbers are derived from these Combinations.
In which case, since the Binary Set equals {0,1}, the total number
of Unique Combinations equals the set {00, 01, 10, 11}, which
respectively represents the first '4' Binary Numbers whose mapping
with the Set of Integers starts with the Number '1'.
Hence, the Correct Method for Enumeration in the Binary System is
given by the Results displayed in Table IIa, and the Modern
Interpretation for the Method of Enumeration in the Binary System is
clearly wrong. But still, both methods clearly represent a Binary
System. Notwithstanding however, while the conclusions derived with
respect to each of these Systems remains unquestionably valid. It
does not stop, nor prevent any decision regarding choice. In other
words, for whatever reason, right or wrong, for now at least, it
does not matter which Binary System is used. Because other than
myself, no one has, or is capable of completing the necessary
studies indicating some out come producing a harm, resulting from
the effects for choosing the wrong System.
E Terrell [Page 34]
Logical Analysis of the Binary System February 13, 2002
References
1. E Terrell ( not published, notarized 1979 ) " The Proof of
Fermat's Last Theorem: The Revolution in Mathematical
Thought" Outlines the significance of the need for a
thorough understanding of the Concept of Quantification
and the Concept of the Common Coefficient. These
principles, as well many others, were found to maintain
an unyielding importance in the Logical Analysis of
Exponential Equations in Number Theory.
2. E. Terrell ( not published, notarized 1983 ) " The Rudiments
of Finite Algebra: The Results of Quantification
" Demonstrates the use of the Exponent in Logical
Analysis, not only of the Pure Arithmetic Functions
of Number Theory, but Pure Logic as well. Where the
Exponent was utilized in the Logical Expansion of the
underlying concepts of Set Theory and the Field
Postulates. The results yield; another Distributive
Property (i.e. Distributive Law for Exponential Functions)
and emphasized the possibility of an Alternate View of the
Entire Mathematical field.
3. G Boole ( Dover publication, 1958 ) "An Investigation of The
Laws of Thought" On which is founded The Mathematical
Theories of Logic and Probabilities; and the Logic of
Computer Mathematics.
4. R Carnap ( University of Chicago Press, 1947 / 1958 )
"Meaning and Necessity" A study in Semantics and
Modal Logic.
5. R Carnap ( Dover Publications, 1958 ) " Introduction to
Symbolic Logic and its Applications"
Author
Eugene Terrell
24409 Soto Road Apt. 7
Hayward, CA. 94544-1438
Voice: 510-537-2390
E-Mail: eterrell00@netzero.net
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Logical Analysis of the Binary System February 13, 2002