Cantor–Dedekind axiom
Imagine you have a line, and you want to talk about all the numbers on that line. Some of these numbers are whole numbers like 1, 2, 3, while others are not whole numbers like 1.5 or 2.8. The Cantor-Dedekind Axiom helps us understand how to group and compare these numbers on a line.
The first part of the axiom is from Dedekind, who said that any real number line can be divided into two sets called the upper and lower sets. The upper set contains all the numbers that are greater than or equal to a specific number on the line, while the lower set contains all the numbers that are less than or equal to that specific number.
Now, let's say we have two numbers on this line, and we want to decide which one is smaller. The Cantor part of the axiom says that we can always find a third number that is in between these two numbers. For example, if we have 1 and 2, we can always find a number like 1.5 in between them. This helps us compare any two numbers on the line and say which one is bigger or smaller.
So, the Cantor-Dedekind Axiom helps us understand how to group and compare numbers on a line. It says we can divide the line into upper and lower sets, and we can always find a number in between any two numbers on the line.
The first part of the axiom is from Dedekind, who said that any real number line can be divided into two sets called the upper and lower sets. The upper set contains all the numbers that are greater than or equal to a specific number on the line, while the lower set contains all the numbers that are less than or equal to that specific number.
Now, let's say we have two numbers on this line, and we want to decide which one is smaller. The Cantor part of the axiom says that we can always find a third number that is in between these two numbers. For example, if we have 1 and 2, we can always find a number like 1.5 in between them. This helps us compare any two numbers on the line and say which one is bigger or smaller.
So, the Cantor-Dedekind Axiom helps us understand how to group and compare numbers on a line. It says we can divide the line into upper and lower sets, and we can always find a number in between any two numbers on the line.