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.
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[2] 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.
Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.
In social choice theory, Arrow's impossibility theorem shows that it is impossible to devise a ranked-choice voting system that is both non-dictatorial and satisfies a basic requirement for rational behavior called independence of irrelevant alternatives.
Gibbard's theorem shows that any strategyproof game form (i.e. one with a dominant strategy) with more than two outcomes is dictatorial.
The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers.There is a famous passage in Plato's Theaetetus in which it is stated that Theodorus (Plato's teacher) proved the irrationality of taking all the separate cases up to the root of 17 square feet ... .
The compass allows a geometer to construct points equidistant from each other, which in Euclidean space are equivalent to implicitly calculations of square roots.
But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots.
Nagel and Newman argue that this may be because the postulate concerns "infinitely remote" regions of space; in particular, parallel lines are defined as not meeting even "at infinity", in contrast to asymptotes.
[12] This perceived lack of self-evidence led to the question of whether it might be proven from the other Euclidean axioms and postulates.
It was only in the nineteenth century that the impossibility of deducing the parallel postulate from the others was demonstrated in the works of Gauss, Bolyai, Lobachevsky, and Riemann.
These works showed that the parallel postulate can moreover be replaced by alternatives, leading to non-Euclidean geometries.
Nagel and Newman consider the question raised by the parallel postulate to be "...perhaps the most significant development in its long-range effects upon subsequent mathematical history".
[14] This profound paradox presented by Jules Richard in 1905 informed the work of Kurt Gödel[15] and Alan Turing.
Franzén's discussion is significantly more complicated than Beltrami's and delves into Ω—Gregory Chaitin's so-called "halting probability".
Chaitin has written a number of books about his endeavors and the subsequent philosophic and mathematical fallout from them.
While most strings are random, no particular one can be proved so, except for finitely many short ones: Beltrami observes that "Chaitin's proof is related to a paradox posed by Oxford librarian G. Berry early in the twentieth century that asks for 'the smallest positive integer that cannot be defined by an English sentence with fewer than 1000 characters.'
"[22] In natural science, impossibility theorems are derived as mathematical results proven within well-established scientific theories.
Another is the uncertainty principle of quantum mechanics, which asserts the impossibility of simultaneously knowing both the position and the momentum of a particle.
There is also Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample.