Proof by infinite descent

The principle of infinite descent is a special mathematical proof method based on the principle of proof by contradiction. This makes use of that there can be no infinite sequence of decreasing numbers in the set of natural numbers, which is equivalent to the fact that every non-empty set of natural numbers has a smallest element. The principle of infinite descent is typically used to show that a given equation has no solution.

Origin

The method of infinite descent was developed in the 17th century by Pierre de Fermat. He used the principle to prove some of his mathematical results. Among others, the special case of Fermat's great theorem was proved using this method.

General Procedure

The task is to prove that a given mathematical problem has no solution in the natural numbers. The proof will now start with the assumption of the existence of a solution. From this solution, constructed using the properties of the natural numbers and the problem an even smaller solution. This process can repeat it by now emanating from the newly found smaller solution, and the result is always smaller solutions in natural numbers. Meaning, it is an infinite, descending sequence of natural numbers, but can not give it, because below a natural number are only finitely many more. This contradiction shows that it was considered a false assumption. However, the only assumption made was the existence of a solution. Thus, this is the only possible source of error. Consequently there is no solution to this problem.

Comparison with the induction principle

The proof of principle of induction is equivalent to the statement that every non-empty set of natural numbers has a smallest element. The latter is equivalent to the statement that there can be no infinite sequences of decreasing natural numbers:

If there are no infinite, descending sequences in the natural numbers, then every non-empty subset has a least element. Had namely a non-empty subset which contains no smallest element, so you could still find on each element of this subset of a smaller and so construct an infinite descending sequence.

Conversely, if every non-empty set of natural numbers has a smallest element, so there can be no infinite, strictly decreasing sequence of natural numbers, because the amount of the followers of such a sequence could have no smallest element.

Therefore, the principle of infinite descent is also based on the fact that every non-empty set of natural numbers has a smallest element.

Example

• The square root of 2 is irrational

Proof: We assume that the square root of 2 is rational.

This means ( as certainly is positive), that there are natural numbers. We show that there are an even smaller, for every, for which there is a with the same property.

Because is certain, that is a natural number. Also, because certain and thus a natural number. From also still follows.

Now one expects after using from: and therefore. So in fact is a natural number, for which there is a natural number.

The principle of infinite descent thus leads to the desired contradiction. Consequently, the square root of 2 is irrational.

Swell

• " The ABC of clear thinking: 22 Thinking Tools for a better life " by Christian Hesse, publisher: Beck; Edition: 2 ( 14 May 2009)
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