Cardinal numbers (0, 1, 2, . . .) are including 0 (zero).
Ordinal numbers (1, 2, 3, . . .) are starting with 1 (first). There is no
"zeroth".
There is arithmetic of cardinal numbers (including the J verbs + * ^ ! ) , but
there is no arithmetic of ordinal numbers.
The codes of the UDC are important numerical objects, but they are neither
integers, nor decimal fractions, nor rational numbers, nor real numbers, nor
complex numbers, nor quaternions, nor vectors, nor matrices, nor functions, nor
operators. They have been neglected by mathematicians.
A new kind of numbers must be considered. I dubbed them 'ordinal fractions' .
A cardinal number, such as 'one', counts a set.
An ordinal number, such as 'the first', identifies an element in a set.
A cardinal fraction, such as 'one half', measures a part of a totality.
An ordinal fraction, such as 'the first half', identifies a part of a totality.
Consider for simplicity the binary, rather than the decimal, notation.
one = 1 = 0001. The digit positions, right to left, indicate ones, twos, fours,
and eights, and the digit values are one-digit binary cardinal numbers, 0 and 1.
the first = 1 = 0001. This is the cardinal number corresponding to the ordinal
number in question.
one half = 0.1 = 0.1000. The digit positions after the binary point indicate
halfs, fourths, eights, and sixteenths, and the digit values are one-digit
binary cardinal numbers, 0 and 1.
the first half = ?????
My solution to this problem is
the first half = 1 = 1000
the second half = 2 = 2000
the first fourth = 11 = 1100
the second fourth = 12 = 1200
the third fourth = 21 = 2100
the fourth fourth = 22 = 2200
the odd fourths = 01 = 0100
the even fourths = 02 = 0200
the sixteenth sixteenth = 2222
Note that the digit positions indicate halfs, fourths, eights, and sixteenths,
and the digit values are either 1 meaning first, and 2 meaning second, or 0
meaning both. 1000 means: first half, both fourths, both eights, both
sixteenths. 'both' goes without saying, just as 0 goes without saying. 1000 = 1
= first half. That is one reason for choosing 0 for 'both'.
I did not know the words hyponymy and hypernymy. Thanks! That is just what I
need.
In logic I let 1 and 2 represent True and False. 0 means unknown or unimportant.
The Transylvanian problem:
0001 Minna is human
0002 Minna is vampire
0010 Minna is sane
0020 Minna is insane
0100 Lucy is human
0200 Lucy is vampire
1000 Lucy is sane
2000 Lucy is insane
Check the 2^4 ordinal fractions from 1111 to 2222 against the data. (I have not
done it)
Thanks! Bo. .
Den 5:58 fredag den 8. juni 2018 skrev Donna Y <dy...@sympatico.ca>:
Can we agree on definitions
Ordinal numbers and Cardinal numbers are Natural numbers which do no include
the number 0. The Natural numbers are well ordered.
Whole numbers are the Natural numbers and 0. 0 is the least element of the
Whole numbers.
Integers are Whole numbers and Negative signed Natural numbers. (-n + n= 0
Zero is not a positive or a negative integer—0 has no sign.) The set of
integers has no least element.
Real numbers are continuous and complete and can be rational or
irrational—rational numbers are integers and fractions, irrational numbers
cannot be expressed as a ratio of two integers.
Imaginary numbers are not real—they exist in another dimension—root negative 1
or i, complex numbers have real and imaginary components.
Tables are two dimensional arrays.
The basic structure of UDC is hierarchy but it could be viewed in other ways—as
you said yourself.
Are you using tables where Rows are records and Columns are attributes? Is
there a primary key?
I am not sure where your Ordinal Fraction comes in but a computer application
of UDC would need full integration of information retrieval (IR) features into
a database management system (DBMS).
Right truncation specifies hypernomy (is a—as in number theory is a subset of
mathematics)—I copied this table but from my view forms of higher degree is not
a subset of diophantine equations.
Table 1:
5: mathematics and natural sciences
51: mathematics
511: number theory
511.5: diophantine equations
511.57: forms of higher degree
What is the precise reason you chose to use 0 as a wild card?—why not * or # or
& or … ? What advantages are derived by using 0? For example when you use
boolean logic the 0 and 1 can represent False and True and then the result
vector of 0 and 1 can be used for selection.
hat does this even mean?
> For example: 0=0 and 0>1 and 1<0 and 10<>01 and 1><2. (meaning that the whole
> is equal to the whole, and the whole comprises the first part, and the first
> part is part of the whole, and the first half is compatible with the odd
> fourths, and the first part is disjoint with the second part).
> The notation for ordinal fractions makes ordinal fraction arithmetic easy,
> just as the notation for cardinal numbers makes cardinal number arithmetic
> easy.
> Keep asking!
Maybe you can show how Ordinal Fractions can be applied to the problem below:
PUZZLE BREAK
Inspector Craig Visits Transylvania
Inspector Craig of Scotland Yard was called to Transylvania to solve some cases
of vampirism. Arriving there, he found the country inhabited both by vampires
and humans. Vampires always lie and humans always tell the truth.
However, half the inhabitants, both human and vampire, are insane and totally
deluded in their beliefs: all true propositions they believe false, and all
false propositions they believe true.
The other half of the inhabitants are completely sane: all true statements they
know to be true, and all false statements they know to be false. Thus sane
humans and insane vampires make only true statements; insane humans and sane
vampires make only false statements.
Inspector Craig met two sisters, Lucy and Minna. He knew that one was a vampire
and one was a human, but knew nothing about the sanity of either.
Here is the investigation:
Craig (to Lucy): Tell me about yourselves.
Lucy: We are both insane.
Craig (to Minna): Is that true?
Minna: Of course not!
From this, Craig was able to prove which of the sisters was the vampire. Which
one was it?
— From Logician Raymond Smullyan
Donna Y
dy...@sympatico.ca
> On Jun 7, 2018, at 5:55 PM, 'Bo Jacoby' via Chat <c...@jsoftware.com> wrote:
>
>
> "You do not show how to access particular rows or columns or elements from a
> table".
> The table, and the table name, is addressed by 00.
> The left column, and the left column header, is addressed by 01.
> The right column, and the right column header, is addressed by 02.
> The upper row, and the upper row header, is addressed by 10.
> The upper left table entry, and its data content, is addressed by 11.
> The upper right table entry, and its data content, is addressed by 12.
> The lower row, and the lower row header, is addressed by 20.
> The lower left table entry, and its data content, is addressed by 21.
> The lower right table entry, and its data content, is addressed by 22.
>
> "without knowing the number of rows or columns."
> In the example there are two rows and two columns.
> If you need a third row, call it 30.
>
> "Why not the concept of arrays with an index?"
> Arrays may have different shapes. Any ordinal fraction has the shape (_$9).
> Arrays have elements. Ordinal fractions don't.
>
> Only array elements contains data. Any ordinal fraction may contain data.
>
> Arrays may have subarrays. Any ordinal fraction has subordinate ordinal
> fractions.
> Arrays and atoms have names. Ordinal fractions don't.
>
> "the set of Natural numbers begin with 1 yet we use base 10 representation
> that uses 0."
>
> Cardinal numbers and ordinal fractions have similarities and differences.
> Cardinal number 0 (meaning "nothing") is not the same thing as ordinal
> fraction 0 (meaning "everything").
>
> Cardinal number 1 (meaning "one") is not the same thing as ordinal fraction 1
> (meaning "first part").
>
> Using "0" for wild card character does not get along with using "0" for
> counting to ten.
> A cardinal number is represented by a right-justified sequence of digits with
> a finite number of nonzero digits. The cardinal number 1 may be written
> 00001.
>
> An ordinal fraction is represented by a left-justified sequence of digits
> with a finite number of nonzero digits. The ordinal fraction 1 may be written
> 10000.
>
> "However this was not an easy system in which to do arithmetic so I cannot
> see how your base 9 system could be either."
> Cardinal numbers, A and B, are ordered such that either A=B or A<B or A>B.
> For example: 0=0 and 0<1 and 1>0 (meaning "zero is equal to zero" , and "zero
> is fewer than one", and "one is more than zero").
>
> Ordinal fractions, A and B, are ordered such that either A=B or A<B or A>B or
> A<>B or A><B. For example: 0=0 and 0>1 and 1<0 and 10<>01 and 1><2. (meaning
> that the whole is equal to the whole, and the whole comprises the first part,
> and the first part is part of the whole, and the first half is compatible
> with the odd fourths, and the first part is disjoint with the second part).
> The notation for ordinal fractions makes ordinal fraction arithmetic easy,
> just as the notation for cardinal numbers makes cardinal number arithmetic
> easy.
> Keep asking!
> Bo.
>
>
>
>
>
>
>
>
>
>
>
>
>
> Den 18:35 torsdag den 7. juni 2018 skrev Donna Y <dy...@sympatico.ca>:
>
>
> UDC is a library classification system akin to the Dewey decimal system—the
> creators collaborated with Dewy. UTC is also used for data.
>
>> Every number is thought of as a decimal fraction with the initial decimal
>> point omitted, which determines the filing order.
>
> I am not sure why you say it cannot handle tables.
>
>> Concepts are organized in two kinds of tables in UDC:[27]
>> <https://en.wikipedia.org/wiki/Universal_Decimal_Classification#cite_note-UDC_Structure-27>
>> Common auxiliary tables (including certain auxiliary signs). These tables
>> contain facets of concepts representing, general recurrent characteristics,
>> applicable over a range of subjects throughout the main tables, including
>> notions such as place, language of the text and physical form of the
>> document, which may occur in almost any subject. UDC numbers from these
>> tables, called common auxiliaries are simply added at the end of the number
>> for the subject taken from the main tables. There are over 15,000 of common
>> auxiliaries in UDC.
>> The main tables or main schedules containing the various disciplines and
>> branches of knowledge, arranged in 9 main classes, numbered from 0 to 9
>> (with class 4 being vacant). At the beginning of each class there are also
>> series of special auxiliaries, which express aspects that are recurrent
>> within this specific class. Main tables in UDC contain more than 60,000
>> subdivisions.
>
> Can you show exactly what is gained by your “ordinal fraction”?
>
> Also—the set of Natural numbers begin with 1 yet we use base 10
> representation that uses 0.
>
> However in the 4 BCE the Greeks used their letters to represent numbers from
> one to nine with no zero:
>
> alpha beta gamma delta epsilon digamma zeta eta theta
>
> They didn’t need to use 0 for higher values because much like Romans they had
> symbols for higher values.
>
> However this was not an easy system in which to do arithmetic so I cannot see
> how your base 9 system could be either.
>
> You suddenly use 0 when you illustrate tables. You do not show how to access
> particular rows or columns or elements from a table but instead use vague
> terms upper, lower, left, right without knowing the number of rows or
> columns..
>
> Why not the concept of arrays with an index?.
>
> Maybe you could explain what I am missing.
>
> Donna Y
> dy...@sympatico.ca
>
>
>> On Jun 7, 2018, at 3:47 AM, 'Bo Jacoby' via Chat <c...@jsoftware.com> wrote:
>>
>> Ordinal Fractions is an improvement to the idea behind the Universal Decimal
>> Classification (UDC).
>
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