Math from Words
Chapter Three - Ordered Sets
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Procedure

Sets p has a one to one correspondence with set q if and only if that determination is made by the following procedure:

Initially, sets r and s are both empty.

AA. If p is not empty and q is not empty, a single element is moved from set p to set r, and a single element is moved from set q to set s, and AA is to be repeated; otherwise, go to BB.

BB. If both p and q are empty, p becomes the union of p and r, q becomes the union of q and s, set p has a one to one correspondence with set q, and this procedure is ended; otherwise (one set is not empty), go to CC.

CC. Set p does not have a one-to-one correspondence with set q, and this procedure is ended.

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The only purpose of sets r and s in the procedure is to restore the contents of set p and set q.

Order

Many of our experiences with the idea of order require quantitative comparison. Used pencils can be arranged in order of length. We can use a ruler, or if the differences are visually obvious, we can make judgements of more than and less than. In either case, we are making quantitative comparisons. Distance, weight, time, brightness, and a great many other quantities can be ordered; but finding a way to define order for mathematical purposes cannot involve the use of any skill in judging quantities. We need to know what the meaning of order is before we define ways to compare quantities. This can't be allowed to be circular. Counting and calculating depend on the concept of order. A definition of order cannot depend on counting and calculating.

The current task related to order is to find a way to define ordered sets. The path taken on this journey involves the concept of association.

There are two cases to consider. Here is case 1: The people of a particular village are accustomed to frequently seeing Guinevere and Reginald together in public places. Guinevere is associated with Reginald, and Reginald is associated with Guinevere. Here is case 2: The children of a school class have learned to effortlessly recite the English alphabet from A to Z, but when asked to recite this alphabet backwards, they all find it difficult to do. Saying L, M, N, O, P is easy. Saying P, O, N, M, L, is a whole other thing to practice. This capability does not come automatically as a result of memorizing the alphabet in the usual way.

Associations that go only in one direction are very common. The Black Sea may remind you of water. The Black Sea is surely associated with water; but water will not reliably remind you of the Black Sea. Water is far from uniquely associated with the Black Sea. The association is not symmetrical. It is asymmetrical.

Remembering the alphabet is remembering a list of asymmetrical associations. Remembering the alphabet backwards is remembering another list of asymmetrical associations.

This is the notation for the symmetrical association:   G ←→ R

This is the notation for an asymmetrical association:   D → E

Since English is read from left to right, the right arrow, rather than the left, is chosen for the asymmetrical association.

Definition - A symmetrical association, G ←→ R, associates R with G and associates G with R.

Definition - An asymmetrical association, X → Y, associates Y with X. In such a case, Y is asymmetrically associated with X.

Definition - X goes to Y, or Y is next in order from X if and only if Y is asymmetrically associated with X.

Definition - V is a text item if and only if each following statement is true:

V is a single character or mark, or V is comprised of adjacent characters or marks written and read horizontally, from left to right.

Each character or mark y, in V, is asymmetrically associated with some character or mark x, in V, and is next in order from that x, in V, except that a single character or mark, h, in V, is not next in order from any character or mark in V.

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Definition - Hereinafter, the terms character and mark shall be synonymous.

Definition - H is the rightmost character in text item V if and only if each following statement is true:

V is a text item.

H is a character in V.

H is the character in V that is located at the right hand end of V.

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Definition - H is the leftmost character in text item V if and only if each following statement is true:

V is an text item.

H is a character in V.

H is the character in V that is located at the left hand end of V.

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Definition - X is the text item that results when C is appended to V on the right if and only if each following statement is true:

V is a text item.

C is a character that is placed just to the right of V, making it the rightmost character in the text item.

X is the text item whose last character is C, and whose other characters are exactly the characters of V, and in the same order.

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Definition - X is the text item that results when X is the text item that results when C is appended to V on the left if and only if each following statement is true:

V is a text item.

C is a character that is placed just to the left of V, making it the leftmost character in text item.

X is the text item whose first character is C, and whose other characters are exactly the characters of V, and in the same order.

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In a culture of imprecise language (as all cultures seem to be) the word joy sometimes refers to a feeling, and sometimes refers to written or read entity that has a spelling formed by a series of marks or characters. The context in which the term word is used often avoids ambiguity, but where new terms are being defined, this particular kind of blurring is difficult to avoid without serious vigilance. A text item might casually be used to refer to an amount because it is associated with a number, but it is vitally important to regard any text item as only the spelling of a word that means a number. The French spells 56 one way and the Germans spell it another way, but people in both countries have the same concept of one and zero and fifty six. A text item represents a number in a set, and a number names an amount in the world.

Definition - S is an ordered set if and only if each following statement is true:

S is a set.

Each element y, in S, is asymmetrically associated with some element x, in S, and is next in order from that x, in S, except that a single element, h, in S is not next in order from any element in S. Such an element as h shall be recognized as The First Element in S.

If element q, in S, is next in order from p, in S, then q is not next in order from any element other than p, in S.

If q, in S, is next in oder from p, in S, then no element other than q is next in order from p.

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Notation - This is the notation for an ordered set, S, containing elements p, q, and r, and written left to right S = (p, q, r).

Definition - An unordered set, U, is a set, such that if p is an element in U, then p is not asymmetrically associated with any element in U, and p is not next in order from any element in U.

The English Alphabet is an example of an ordered set.

In this development, the terms letter, alphabet, English alphabet, and alphabetical order are undefined terms regarded as empirical terms, learned from experience. Unless otherwise stated, an alphabet contains only English characters, and does not include punctuation or characters other that those in the following pair of ordered sets:

The Upper-Case Set of English Letters = (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z), and the Lower-Case Set of English Letters = (a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z).

Names may include Greek letters such as these:

α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π ρ σ τ υ φ χ ψ ω

Some of the more often seen and easily pronounced Greek letters are these:

α = alpha, β = beta, γ = gamma, δ = delta, ε = epsilon, θ = theta, ι = iota, λ = lambda, μ = mu, ν = nu, π = pi, σ = sigma, and ω = omega.

Definition - S is a bounded ordered set if and only if the statements in any of the following cases are true:

Case I

S contains no elements, and S is an empty set.

Case II

S is a set containing only a single element, t, and t is both the first and the last element in S.

Case III

S is a set.

Each element y, in S, is asymmetrically associated with some element x, in S, and is next in order from that x, in S, except that a single element, h, in S is not next in order from any element in S. Such an element as h shall be recognized as The First Element in S.

Element j, in S, is such that no element in S is next in order from j. Such an element as j shall be recognized as The Last Element in S

If element q, in S, is next in order from p, in S, then q is not next in order from any element other than p, in S.

If q, in S, is next in order from p, in S, then no element other than q is next in order from p.

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Definition - S is a bounded subset of ordered set K if and only if each following statement is true:

S is a set.

Each element in S is an element in K.

S is a proper subset of K.

Each element y, in S, is asymmetrically associated with some element x, in S, and is next in order from that x, in S, except that a single element, h, in S is not next in order from any element in S. Such an element as h shall be recognized as The First Element in S.

Element j, in S, is such that no element in S is next in order from j. Such an element as j shall be recognized as The Last Element in S

If element q, in S, is next in order from p, in S, then q is not next in order from any element other than p, in S.

If q, in S, is next in oder from p, in S, then no element other than q is next in order from p.

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Definition - W is the set of whole numbers if and only if each following statement is true:

W is an ordered set.

Each element in W is a number naming an amount of things.

The amount of elements in any set of things is named by a number which is an element in W.

If a set S is empty, the amount of things in S is named by the first number in set W.

If set S contains only a single thing, the amount of things in S is named by the number in W that is next in order from the first.

Each element in W is represented in writing by a text item.

No two text items are identical that represent two different elements respectively, in set W.

The first element in W is the number zero, the name of the amount of a things in an empty set, or the word none.

The text item representing the first element in W is 0.

The element next in order from zero, in W, is the number one, the name of the amount of a single thing.

The text element representing one, in W, is 1.

Whenever the amount of things in a set S is named by a number r, in W, and subsequently an element is inserted in S, then the amount of things then in S is a single thing greater than it was previously, and this new amount is named by the number next in order from r, in W.

The amount of individually separate, distinct and discrete things, however many, in any set of such things can be named by a number in W, and represented by a unique text item representing that number in W.

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Definition - S is an unbounded ordered set if and only if each following statement is true:

S is a set.

Each element y, in S, is asymmetrically associated with some element x, in S, and is next in order from that x, in S, except that a single element, h, in S is not next in order from any element in S. Such an element as h shall be recognized as The First Element in S.

If element q, in S, is next in order from p, in S, then q is not next in order from any element other than p, in S.

If q, in S, is next in oder from p, in S, then no element other than q is next in order from p.

The amount of individually separate, distinct and discrete things, however many, in any set of such things can be named by a number in W, and represented by a unique text item representing that number in W.

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Notice that the set of whole numbers, W, is an unbounded set.

Now suppose c1 and c2 are amounts of elements in two sets, and x and y are the whole numbers which name c1 and c2. Suppose also that y is next in order from x in the set of whole numbers and that c1 is greater than c2 by 1. In a new manner of speaking, we simply say that the whole number x is greater than the whole number y, by 1.

While the following definitions will not define addition or subtraction in general, or even for all pairs of whole numbers, they will be useful soon in this development.

Definition - Whole number 1 added to whole number x is equal to y, or y is one greater than x if and only if each following statement is true:

W is the set of whole numbers.

Elements of W include 1, x, and y.

Number y is next in order from x in W.

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Notation - The addition of 1 to any whole number x is notated this way:

x + 1 = y

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we can also observe that x is 1 less than y.

Definition - Whole number 1 subtracted from whole number y is equal to x, or x is one less than y if and only if each following statement is true:

W is the set of whole numbers.

Elements of W include 1, x, and y.

Number y is next in order from x in W.

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Notation - The subtraction of 1 from any whole number y is notated this way:

y - 1 = x

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Definition - Whole number p is the name of the amount of elements in a set S if and only if the set S has a one-to-one correspondence with the ordered set of whole numbers from 1 to p, as established with the following procedure.

Procedure

L0. W is the set of whole numbers.

L1. Set σ (sigma) is empty;

L2. Whole number x is 0;

L3. If set S is empty go to L7;

L4. An element e is moved from set S to set σ ;

L5. The whole number that is next in order from x, in W, is now the whole number that x becomes;

L6. Go to L3;

L7. Each element in σ is moved to set S.

This last instruction restores S to its original contents.

Each iteration of L4 and L5 associates a newly removed element e from S with the next in order element x in W. The correspondence required by the definition is established, AND p is the amount of elements in set S, if and only if x is p.

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