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Proof of impossibility
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Proof of impossibility
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution by proving there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.
The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.
Some of the most important proofs of impossibility found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm, with one of the more prominent ones being the halting problem. Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.
In computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques that are not affected by relativization cannot resolve the P versus NP problem. Another technique is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by showing them to be just as hard to solve as any other problem in the class. In particular, a complete problem is intractable if one of the problems in its class is.
One of the widely used types of impossibility proof is proof by contradiction. In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive. Since the contradiction stems from the original assumption, this means that the assumed premise must be impossible.
In contrast, a non-constructive proof of an impossibility claim would proceed by showing it is logically contradictory for all possible counterexamples to be invalid: at least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture. For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved, by showing that one of two possible counterexamples must be a valid counterexample, without showing which one it is.
Another type of proof by contradiction is proof by descent, which proceeds first by assuming that something is possible, such as a positive integer solution to a class of equations, and that therefore there must be a smallest solution (by the Well-ordering principle). From the alleged smallest solution, it is then shown that a smaller solution can be found, contradicting the premise that the former solution was the smallest one possible—thereby showing that the original premise that a solution exists must be false.
The obvious way to disprove an impossibility conjecture is by providing a single counterexample. For example, Euler proposed that at least n different nth powers were necessary to sum to yet another nth power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power:
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Proof of impossibility AI simulator
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Proof of impossibility
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as proofs of impossibility, negative proofs, or negative results. Impossibility theorems often resolve decades or centuries of work spent looking for a solution by proving there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.
The irrationality of the square root of 2 is one of the oldest proofs of impossibility. It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.
Some of the most important proofs of impossibility found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm, with one of the more prominent ones being the halting problem. Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.
In computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques that are not affected by relativization cannot resolve the P versus NP problem. Another technique is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by showing them to be just as hard to solve as any other problem in the class. In particular, a complete problem is intractable if one of the problems in its class is.
One of the widely used types of impossibility proof is proof by contradiction. In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive. Since the contradiction stems from the original assumption, this means that the assumed premise must be impossible.
In contrast, a non-constructive proof of an impossibility claim would proceed by showing it is logically contradictory for all possible counterexamples to be invalid: at least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture. For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved, by showing that one of two possible counterexamples must be a valid counterexample, without showing which one it is.
Another type of proof by contradiction is proof by descent, which proceeds first by assuming that something is possible, such as a positive integer solution to a class of equations, and that therefore there must be a smallest solution (by the Well-ordering principle). From the alleged smallest solution, it is then shown that a smaller solution can be found, contradicting the premise that the former solution was the smallest one possible—thereby showing that the original premise that a solution exists must be false.
The obvious way to disprove an impossibility conjecture is by providing a single counterexample. For example, Euler proposed that at least n different nth powers were necessary to sum to yet another nth power. The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: