Elementary Arithmetic - Successorship and Size

Successorship and Size

The result of the addition of one to a number is the successor of that number. Examples:
the successor of zero is one,
the successor of one is two,
the successor of two is three,
the successor of ten is eleven.
Every natural number has a successor.

The predecessor of the successor of a number is the number itself. For example, five is the successor of four therefore four is the predecessor of five. Every natural number except zero has a predecessor.

If a number is the successor of another number, then the first number is said to be larger than the other number. If a number is larger than another number, and if the other number is larger than a third number, then the first number is also larger than the third number. Example: five is larger than four, and four is larger than three, therefore five is larger than three. But six is larger than five, therefore six is also larger than three. But seven is larger than six, therefore seven is also larger than three ... therefore eight is larger than three ... therefore nine is larger than three, etc.

If two non-zero natural numbers are added together, then their sum is larger than either one of them. Example: three plus five equals eight, therefore eight is larger than three (8 > 3) and eight is larger than five (8 > 5). The symbol for "larger than" is >.

If a number is larger than another one, then the other is smaller than the first one. Examples: three is smaller than eight (3 < 8) and five is smaller than eight (5 < 8). The symbol for smaller than is <. A number cannot be at the same time larger and smaller than another number. Neither can a number be at the same time larger than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true:

• the first number is larger than the second one,
• the first number is equal to the second one,
• the first number is smaller than the second one.