See my notes on your explanation of Ordinal Fractions Donna Y [email protected]
> On Jun 8, 2018, at 3:40 AM, 'Bo Jacoby' via Chat <[email protected]> wrote: I am beginning to understand some things you are saying about your concept of Ordinal Fractions. Ideas that are explained and seem elegant, useful, or intuitive, may catch on. I made comments between your assertions that may or not help you refine the presentation of your ideas. you say: > Cardinal numbers (0, 1, 2, . . .) are including 0 (zero). > Ordinal numbers (1, 2, 3, . . .) are starting with 1 (first). There is no > "zeroth”. It is better to use the same least number for the Cardinal and Ordinal numbers. The set of natural numbers and zero is called the whole numbers . The set of whole numbers is usually denoted by the symbol W W={0,1,2,3,4,5,6,…} Cardinal Numbers and Ordinal Numbers are Natural Numbers— Cardinals can be enumerated by ordinals and the two can be put into one-to-one correspondence. Since you chose to use Ordinals {1 2 3…} I chose a definition of Natural numbers beginning with 1 which applies to the Ordinals and Cardinals in that system. I was trying to impose 1 as the first element of both Cardinal and Ordinal numbers as they are both Natural numbers since it seems important to you for your Ordinal Fractions that Ordinal numbers start at 1 Peano tried to define a least number of axioms to derive all of Arithmetic and the Real numbers from the Natural numbers—his definitions and proofs included the following For Natural Numbers: 1 is not the successor of any number. Every number except 1 has a predecessor. The Peano axioms were meant to provide a rigorous foundation for the Natural numbers and provide the foundation to construct the Real numbers. However a decade later Peano included 0 with the Natural Numbers and revised his Axioms The Peano axioms that allow us to add up the world can be expressed as both including or excluding Zero. Zero absolutely exists and the functionality of arithmetic depends on it. However, its existence is actually formally undecidable. For this reason it must be axiomatized. Usually the smallest natural number is given as either 0 or 1. It depends on what you hope to achieve in your subsequent development of number theory. Peano's Axioms 1. Zero is a number. 2. If is a number, the successor of is a number. 3. zero is not the successor of a number. 4. Two numbers of which the successors are equal are themselves equal. 5. (induction axiom.) If a set of numbers contains zero and also the successor of every number in , then every number is in . Ordinals are traditionally denoted by lower case Greek letters α, β, γ, δ, . . . Cardinal numbers represent the total number of a group of objects vs. Ordinal numbers represent the position of an object in a linear sequence. If you count a set of objects you might begin to count at 1 but if you ask how many objects there are the answer is sometimes 0—the answer may also be 0 if you ask how many are left. >> There is arithmetic of cardinal numbers (including the J verbs + * ^ ! ) , >> but there is no arithmetic of ordinal numbers. Finite Ordinals and Finite Cardinals correspond. The propositions of finite Cardinal arithmetic and Ordinal arithmetic correspond point by point. For the infinite, things are different. > 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. The UDC is a classification system using Real numbers, specifically decimal numbers (floating point) that are formated without leading zeros nor decimal separator since UDC have no integers part but only decimal fractions (i.e. expressed in base 10). For ease of reading, the UDC code is usually punctuated after every third digit. An important property of Real decimal fractions is that there is always more real numbers between any two real numbers. This makes Real numbers ideal for the construction of compound numbers to denote sets of interrelated subjects that could never be exhaustively foreseen. 599.74 Carnivora (carnivorans) 599.744 Canidae. Ursidae. Musteloidea 599.744.2 Ursidae 599.744.21 Ursus (genus) 599.744.211 Brown bears (grizzly) 599.744.212 Polar bears The work to UDC is maintaining the schedules that contain the outline of the various disciplines of knowledge, arranged in 10 classes and hierarchically divided and subdivided and sub-subdivided. > > A new kind of numbers must be considered. I dubbed them 'ordinal fractions’ The only thing different is the application—the numbers map to tables or in other words a data base with a catalogue of subjects hierarchically divided. They are a database key. UDC codes are Real Numbers. The advantage of Real Numbers for library science is that you can always find a code for a new book without having to reclassify. If you look at any two real numbers on a line, no matter how close together, you will find an infinite number of other real numbers—rational and irrational—between them. And if you were to choose two numbers from the infinity between your two numbers, you would find that even between these two new numbers, there exists an infinite amount of real numbers. The set of Real numbers--members are infinite in number, and are arranged in order so that between any pair, an infinite number of new members can be intercalated without affecting the order of the original members of the series. ... the principle of abstraction implies the existence of sets the elements of which are all objects having a certain property For instance, {x | x is blue} is the set of all blue objects. This illustrates the fact that the principle of abstraction implies the existence of sets the elements of which are all objects having a certain property. The main table for UDC is MAIN TABLES 0 SCIENCE AND KNOWLEDGE. ORGANIZATION. COMPUTER SCIENCE. INFORMATION. DOCUMENTATION. LIBRARIANSHIP. INSTITUTIONS. PUBLICATIONS 1 PHILOSOPHY. PSYCHOLOGY 2 RELIGION. THEOLOGY 3 SOCIAL SCIENCES 5 MATHEMATICS. NATURAL SCIENCES 6 APPLIED SCIENCES. MEDICINE. TECHNOLOGY 7 THE ARTS. RECREATION. ENTERTAINMENT. SPORT 8 LANGUAGE. LINGUISTICS. LITERATURE 9 GEOGRAPHY. BIOGRAPHY. HISTORY It uses 10 Arabic numbers 0 to 9 In an ordered set, a collection of objects placed in some order, ordinal numbers are labels for the positions of those ordered objects. By the way the term Zeroth was coined by physicists introducing a forth Law of Thermodynamics more fundamental than the three already well known You said: > . > A cardinal number, such as 'one', counts a set. It counts elements of a set. The number of elements in a set is the cardinal number of that set. When counting the elements of a set you can count them in any order—a Cardinal Number represents a tally of the elements. > An ordinal number, such as 'the first', identifies an element in a set. You have introduced a definition so that the the forth Cardinal number is three since your least value Cardinal is 0 but your least value Ordinal is 1. > > 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. Using a radix or base—all this information is there. > 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 = ????? introducing decimal fractions you are no longer speaking of Cardinal nor Ordinal numbers—these are fractions. In English names of Cardinal numbers are one, two, three and Ordinals first, second, third and fractions if cutting into two-- half, quarter, eighths—don’t confuse a partial overlap on naming rational fractions and the names used for ordinals to mean fractions are ordinal numbers. In other languages such as Polish there are different names for these three types of numbers. The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero. The set of integers is sometimes written J or Z for short. The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 13 ⅓ and −1111/8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z/1. Given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 0). An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repe The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. ½ is decimal 0.5 or binary 0.1 and 0.75 decimal is 0.11 binary It is not always so simple—for example, 0.1 in decimal — to 20 bits — is 0.00011001100110011001 in binary since it is rounded off 0.00011001100110011001 in binary is 0.09999942779541015625 in decimal > My solution to this problem is > the first half = 1 = 1000 > the second half = 2 = 2000 the numerals of binary numbers are 0 and 1 .0 .1 .01 .011 100 101 110 0.101 [2] = (1 x 2*-1) + (0 x 2*-2) + (1 x 2*-3) > > 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 I am not clear on this—it seems needless and confusing. Have you actually implemented this and used it—can you describe your application. > 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 UDC 5 or 500 is all of Math, 51 is sum subset etc.—this concept of the first half and the second half is not meaningful for UDC Maybe you have another application in mind ≈ > > 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. > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
