Chapter B

Creating Numbers

Strings

Definition - S is a string in T if and only if the following statements are true:
T is an ordered set.
S is either an ordered set or S is a set containing a single element.
Each element in S is also an element in T.
If x and y are in S, and x is associated with y in T, then x is associated with y in S.
If p is associated with q in T, and p is not is S, and q is in S, then there is no element y in S such that y is associated with q.
If q is the starting element of T, and q is in S, then there is no element y in S such that y is associated with q.
If p is associated with q in T, and q is not in S, and p is in S, then p is not associated with any element in S.
If p is the ending element of T, and p is in S, then p is not associated with any element in S.
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Definition - Element x in S is the starting element of string S if and only if the following statements are true:
T is an ordered set.
S is a string in T.
There is no element in S that is associated with x.
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Definition - Element z in S is the ending element of string S if and only if the following statements are true:
T is an ordered set.
S is a string in T.
Element z in S is not associated with any element in S.
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Notice that S is either an ordered set or S contains only a single element.   S is not empty. If S contains only a single element, then p = q and this element is both the starting element of S and the ending element of S.

Notation - A string S containing only a single element e may be written this way: S = {e}.

Definition - S is the string from x to y in T if and only if T is an ordered set, and S is a string in T, and x is the starting element of S, and y is the ending element of S.
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Concatenation

Definition - C is the concatenation of S and T if and only if the following statements are true:
C is an ordered set.
S is an ordered set, or S is a string, or S is a set containing one and only one element.
T is an ordered set, or T is a string, or T is a set containing one and only one element.
The intersection of S and T is empty.
C is the union of S and T.
If x is associated with y in S then x is associated with y in C, and if j is associated with k in T then j is associated with k in C.
If S and T are ordered sets, then the ending element of S in C is associated with the starting element of T in C.
If S contains only a single element e, and T is an ordered set, then e in C is associated with the starting element of T in C.
If S is an ordered set and T contains only a single element d, then the ending element of S in C is associated with d in C.
If S contains only a single element g, and T contains only a single element, then C is an ordered pair and g is its starting element.
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Notice also that the concatenation of sets X and Y is not the same as the concatenation of Y and X.   In the latter case, the starting element of the concatenation would be the starting element of Y or Y itself, whereas in the former case, the starting element of the concatenation would be the starting element of X or X itself.

Definition - X is concatenated with Y if and only if a set C is the concatenation of X and Y.

One-To-One Correspondence

Definition - Set A has a one-to-one correspondence with set B if and only if either A is empty and B is empty; or every element in A is in B, and every element in B is in A.
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Definition - A and B have a one-to-one correspondence if and only if A has a one-to-one correspondence with B.

Integer sets

Definition - S is an integer set if and only if the following statements are true:
L is an ordered set, and R is an ordered set.
Each element in L and R is not a set.
The intersection of L and R is empty.
L has a one-to-one correspondence with R.
Element q is not a set.
Element q is not in L or R.
C is the concatenation of L and q.
S is the concatenation of C and R.
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Notice that any integer set is an ordered set.

Definition - L is the set of negative integers of integer set S if and only if the following statements are true:
L is an ordered set, and R is an ordered set.
Each element in L and R is not a set.
The intersection of L and R is empty.
L has a one-to-one correspondence with R.
Element q is not a set.
Element q is not in L or R.
C is the concatenation of L and q.
S is the concatenation of C and R.
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Definition - R is the set of positive integers of integer set S if and only if the following statements are true:
L is an ordered set, and R is an ordered set.
Each element in L and R is not a set.
The intersection of L and R is empty.
L has a one-to-one correspondence with R.
Element q is not a set.
Element q is not in L or R.
C is the concatenation of L and q.
S is the concatenation of C and R.
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Definition - Element q in integer set S is the center element of S if and only if q is associated with the starting element of the set of positive integers of S.

Notation - If R = (v, w, x) = (v → w → x), and T = (a, b, c) = (a → b → c), then the integer set S containing center element q is written this way:

S = ( a, b, c, q, v, w, x) = (a → b → c → q → v → w → x)

Integers

Definition - N is a number if and only if N is an element in ordered set S, and each element in S is not a set.
Definition - N is an integer if and only if N is an element contained in an integer set.
Definition - W is the set of whole numbers of integer set S if and only if W is the concatenation of the center element of S and the set of positive integers of S.
Definition - Element w is a whole number if and only if w is in the set of whole numbers of an integer set.
Definition - Element p is a positive integer if and only if p is in the set of positive integers of an integer set.
Definition - Integer p is positive if and only if p is in the set of positive integers of an integer set.
Definition - Element q is a negative integer if and only if q is in the set of negative integers of an integer set.
Definition - Integer q is negative if and only if q is in the set of negative integers of an integer set.
Definition - As stated earlier, a numeral is a single-character symbol used singly or in combinations to form names.
Definition - N is zero or N = zero if and only if N is the center element of an integer set.
Definition - N is one or N = one if and only if N is the starting element of the set of positive integers of an integer set.
Definition - N is two or N = two if and only if N is associated with one in an integer set.

Notation - If N is zero, then N may be named by and known by the numeral 0.
Notation - If N is one, then N may be named by and known by the numeral 1.
Notation - If N is two, then N may be named by and known by the numeral 2.

Other notations are possible for two. These notations and definitions do not preclude other names.

Zero is the center element of an integer set.   An integer set is the result of two concatenations.   What is eventually defined as the negative integers Q of integer set S is concatenated with an element later called zero.   Zero is thereby associated with the ending element of Q.   This concatenation is concatenated with what is later defined as the positive integers of S.

Definition - Elements n and m are symmetric integers if and only if the following statements are true:
Q is an integer set containing integers m and n.
S is a string in Q such that n is the starting element of string S, and the ending element of S is zero.
W is a string in Q such that zero is the starting element of string W, and m is the ending element of string W.
S has a one-to-one correspondence with W.
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Definition - Integer n is minus one if and only if n and one are symmetric.

Notation - If N is minus one, then N may be named by and known as -1.   It might be possible to devise a numeral to stand for minus one, but this is the traditional notation.   It reminds us of the integer with which it is symmetric.

Definition - Integer n is minus two if and only if n and two symmetric.
Definition - Integer y is the negative of integer x if and only if x and y symmetric.
Definition - Integer y is minus n if and only if n and y symmetric.

Notation - If y is minus n, then y may be named by and known as -n.

Set Size

Definition - The size of T equals x if and only if T is an empty set and x = 0, or the following statements are true:
T is a non-empty set.
P is the set of positive integers of integer set Q.
T has a one-to-one correspondence with the string from 1 to x in P.
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Notation - The size of set A is written this way: | A |.

Definition - The size of set R is equal to the size of set T if and only if either R has a one-to-one correspondence with T; or, if x = | R | and y = | T |, then x = y.

Notation - "The size of set R is equal to the size of set T" may be written this way: | R | = | T |.

Definition - The size of set R is less than the size of set T if and only if R is empty and T is not empty, or the following statements are true:
P is the set of positive integers of integer set Q.
S is a string in P such that the starting element of S is the starting element of P.
V is a string in P such that the starting element of V is the starting element of P.
| S | = | R |.
| V | = | T |.
Element x is in P.
S does not contain x.
V contains x.
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Definition - The size of T is greater than the size of R if and only if the size of R is less than the size of T.

Notation - The symbol for less than is < so in order to say "the size of set R is less than the size of set T," we may say | R | < | T |.
Notation - The symbol for greater than is > so in order to say "the size of set R is greater than the size of set T," we may say | R | > | T |.

Definition - T contains N elements if and only if T is a set and the size of T equals N.