Zhi Nuo, an ancient Greek mathematician who was a little later than Pythagoras, once put forward some famous paradoxes, which had an important influence on later concepts of mathematics and physics. Achilles paradox is one of them.

Achilles is a running hero in Greek mythology. Zhi Nuo said: Achilles couldn't catch up with the tortoise who started earlier than him in the race, because when he reached the point where the tortoise started, the tortoise crawled forward again. The distance between Achilles and the tortoise can be narrowed infinitely, but it will never catch up with the tortoise.

Two different time measures are used in Achilles paradox. The general measurement method is: suppose that the distance between Achilles and tortoise is s at first, and the speed is V 1 and V2 respectively. When the time is t = s/(V 1-V2), Achilles catches up with the tortoise.

But Zhi Nuo's measurement method is different: Achilles will reach the starting point before the tortoise one by one, this time t'. For any T', it may be infinitely shortened, but Achilles is always behind the tortoise. The key point is that this t' cannot measure the time after t = s/(V 1-V2).

Dichotomy paradox

This is also a paradox put forward by Zhi Nuo: when an object travels a certain distance to reach D, it must first reach half of the distance D, then a quarter, an eighth and a sixteenth, so that it can be divided indefinitely. Therefore, this object will never reach D.

Zhi Nuo even thought: "It is impossible to move from one place to another, because if there is such a movement, there will be' perfect infinity', which is impossible." If Achilles really catches up with the tortoise at T, then, "this is an illogical phenomenon, so it is by no means the truth, but just a scam". This means that the senses are unreliable and there is no logical reliability.

He believes that "it is absolutely impossible to be endless."

A hammer of one foot will take half of it every day, and it will never end.

This is a famous saying of Hui Shi in The World of Zhuangzi. More than two thousand years ago, the ancients in China also used the concept of infinity.

Hui Shi, a famous scholar of Song Dynasty during the Warring States Period, was once the Prime Minister of Liang, a wizard of argument, a good friend of Zhuangzi, and a representative figure of famous scholars tied with Gong Sunlong.

Hui Shi's theory emphasizes the * * * phase of everything, so the difference between things is only a relative concept. There are some strange propositions related to Hui Shi, such as "Mountain and Zeping", "Egg has hair", "Chicken has three legs", "Dog can be a cow", "Fire is not hot", "Moment is not square", "White dog is black" and "A solitary pony has no mother". Hui Shi's paradox is also very influential in the west.

Mao Zedong said: "Lenin said that everything can be divided. Taking atoms as an example, not only atoms can split, but also electrons can split. " He added: "The electron itself has not split yet, and it will split one day. It is a truth that' a hammer of one foot, take half of it every day, will last forever'. If you don't believe me, just try. If there is exhaustion, there will be no science. "

There are as many points on the line segment of 1 cm as on the Pacific Ocean.

How many philosophers and mathematicians are afraid of falling into paradox. German mathematician Cantor received his doctorate at the age of 23 and declared war on infinity six years later. He successfully proved that the points on the straight line can correspond to the points on the plane one by one, and can also correspond to the points in the space one by one. Because of infinity, there are as many points on the line segment of 1 cm as there are points in the Pacific Ocean and the whole earth.

However, Cantor's "infinite set" conflicted with the traditional mathematical concept and was reviled. It was not until 1897 that his achievements were recognized, and almost all mathematics was based on set theory. Russell praised his work as "probably the greatest work that this era can boast of."

At the same time, there are some self-contradictory phenomena in set theory, especially Russell's Barber Paradox, which impacts the foundation of mathematics in a very concise form, which is the "third mathematical crisis"