Reduction to absurdity is one of the methods of indirect argument. It is a demonstration method to establish the authenticity of the topic by judging the falsity of the judgment that contradicts the topic (that is, the antithesis).

The argumentation process of reduction to absurdity is as follows: firstly, the thesis is put forward; Then set a counter-proposition, reason according to the reasoning rules, and prove the falsity of the counter-proposition; Finally, according to law of excluded middle, since the counter-proposition is false, the original proposition is true.

In disproof, only the judgment that contradicts the proposition can be regarded as a counter-proposition, while the opposite judgment of the proposition can't be regarded as a counter-proposition, because two opposing judgments can be false at the same time. An important part of reduction to absurdity is to determine the falsity of counter-proposition, and reduction to absurdity is often used.

Many propositions about the divisibility of pure numbers can only be proved by reduction to absurdity. Such problems are usually directly used as theorems or common inferences, such as the root number 2 is irrational. Many known problems have only two elements. Because of the limited conditions, basically we can only use reduction to absurdity. This kind of problem usually has only A and B in an axiomatic system, and the truth value of the unknown proposition is deduced from the known proposition.

Prove that a set has infinite elements by reduction to absurdity. In other words, if there is restriction, there will be contradictions; Map with another infinite set, and the added known infinite set appears as lemma.