As mentioned earlier, the trajectory formed by moving zero-dimensional points is a one-dimensional line, and the straight line or curve at this time is a continuous line. Conversely, a continuous one-dimensional space is made up of an infinite number of points.
As mentioned above, Euclid wrote in the Primitive Geometry that "points in a straight line exist evenly", but he did not say anything about what it meant if the points existed in this way.
However, the points on the line have been examined before Euclid. That was Zeno, the philosopher who lived in the ancient Greek colony of Elia in the 5th century BC, which is now the southern part of the Italian peninsula.
He is best known for proposing the famous "Achilles Paradox". This paradox reveals that a continuous, one-dimensional space is made up of an infinite number of points. Zeno makes the following argument in the paradox, proving that the hero Achilles of ancient Greek mythology, known for his speed, could not run the slow-eating turtle.
The Paradox of Achilles and the Tortoise
Achilles ran from behind the turtle in the same direction as the turtle. Suppose Achilles is twice as fast as the turtle, but by the time he reaches the turtle's position, the turtle is already running ahead (1/2 of the way Achilles ran). When Achilles arrived at that location, the tortoise was still 1/2 ahead of Achilles. The distance between Achilles and the turtle is shortened to 1/2, 1/4, 1/8, 1/16 of the initial distance... But the distance (line) between the two can be infinitely divided, and by repeating it indefinitely, Achilles will never catch up with the turtle—the premise of the paradox that seems unclear at first glance is that a one-dimensional line consists of infinitely many points.
About 2,000 years after Zeno, the understanding of one dimension has reached a deeper level. The 19th-century German mathematician Richard Deiderkin mathematically proved the fact that the number of points formed by cutting a line into two segments cannot completely fill the one-dimensional space, that is, even if it is infinitely divided, it is not enough to form the "continuity" necessary for one-dimensional lines.
The "numbers "numbers", which we use in counting and measuring, have been regarded as one-dimensional straight lines since ancient times. Like a thermometer, mark a line with a scale number (called a "number axis") to the ruler around you, so that the marked integers are scattered. But if there are values like 1/2 or 1/3 that can be represented by integer ratios (rational numbers), there are infinite numbers on the number axis, because both 617/2839 and 850325/1048576 are rational numbers.
As Zeno explained, there are innumerable rational numbers between two values, no matter how small the difference between them, so they are continuous. But Dedekin came up with a "dividing line" approach to disprove this. The investigation of the existence of values on both sides of the breakpoint after the straight line segmentation is called "Dedekin segmentation".
Dedekin split
If you use this method, you can find that one of the intercepts must be irrational, even if it cannot be represented by integer ratios. By adding irrational numbers to rational numbers in a one-dimensional axis, the integrity of real numbers can be achieved for the first time. In other words, this indicates that all real points exist on the number axis.