A Study from Wikipedia

1. Impossible?
You Need a Proof Please.
Dr. Moloy De

2. Proof of impossibility
A proof of impossibility, also known as negative proof, proof of
an impossibility theorem, or negative result, is a proof demonstrating that
a particular problem cannot be solved, or cannot be solved in general.
Often proofs of impossibility have put to rest decades or centuries of work
attempting to find a solution.
To prove that something is impossible is usually much harder than the
opposite task; it is necessary to develop a theory.
Impossibility theorems are usually expressible as universal propositions in
logic (see universal quantification).

3. Proof of impossibility
One of the most famous proofs of impossibility was the 1882 proof
of Ferdinand von Lindemann, showing that the ancient problem of squaring
the circle cannot be solved, because the number π is transcendental (non-
algebraic) and 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 nineteenth century.

4. Proof of impossibility
A problem arising in the sixteenth century was that of creating a general
formula using radicals expressing the solution of any polynomial equation of
fixed degree k, where k ≥ 5.
In the 1820s, the Abel–Ruffini theorem showed this to be impossible using
concepts such as solvable groups from Galois theory, a new subfield
of abstract algebra.
Among the most important proofs of impossibility of the 20th century, were
those related to undecidability, which showed that there are problems that
cannot be solved in general by any algorithm at all. The most famous is
the halting problem.

5. Proof of impossibility
In computational complexity theory, techniques like relativization (see oracle
machine) provide "weak" proofs of impossibility excluding certain proof
techniques.
Other techniques like proofs of completeness for a complexity class provide
evidence for the difficulty of problems by showing them to be just as hard to
solve as other known problems that have proved intractable.

6. Proof by contradiction
One widely used type of impossibility proof is proof by contradiction.
In this type of proof it is shown that if something, such as a solution to a
particular class of equations, were possible, then two mutually contradictory
things would be true, such as a number being both even and odd.
The contradiction implies that the original premise is impossible.

7. Proof by descent
One type of proof by contradiction is proof by descent.
Here it is postulated that something is possible, such as a solution to a class
of equations, and that therefore there must be a smallest solution; then
starting from the allegedly smallest solution, it is shown that a smaller
solution can be found, contradicting the premise that the former solution was
the smallest one possible.
Thus the premise that a solution exists must be false.

8. Proof by descent
This method of proof can also be interpreted slightly differently, as the
method of infinite descent.
One postulates that a positive integer solution exists, whether or not it is the
smallest one, and one shows that based on this solution a smaller solution
must exist.
But by mathematical induction it follows that a still smaller solution must
exist, then a yet smaller one, and so on for an infinite number of steps.
But this contradicts the fact that one cannot find smaller and smaller positive
integers indefinitely; the contradiction implies that the premise that a solution
exists is wrong.

9. Types of disproof of impossibility
conjectures
There are two alternative methods of proving wrong a conjecture that
something is impossible: by counterexample (constructive proof) and by
logical contradiction (non-constructive proof).
The obvious way to disprove an impossibility conjecture 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:
275 + 845 + 1105 + 1335 = 1445

10. Types of disproof of impossibility
conjectures
A proof by counterexample is a constructive proof.
In contrast, a non-constructive proof that something is not impossible
proceeds 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.

11. The Pythagoreans' proof
The proof by Pythagoras (or more likely one of his students) about
500 BCE has had a profound effect on mathematics.
It shows that the square root of 2 cannot be expressed as the ratio of two
integers (counting numbers).
The proof bifurcated "the numbers" into two non-overlapping collections—
the rational numbers and the irrational numbers.
This bifurcation was used by Cantor in his diagonal method, which in turn was
used by Turing in his proof that the Entscheidungsproblem (the decision
problem of Hilbert) is undecidable.

12. The Pythagoreans' proof
It is unknown when, or by whom, the "theorem of Pythagoras" was
discovered. The discovery can hardly have been made by Pythagoras himself,
but it was certainly made in his school. Pythagoras lived about 570–490 BCE.
Democritus, born about 470 BCE, wrote on irrational lines and solids ...
— Heath

13. The Pythagoreans' proof
Proofs followed for various square roots of the primes up to 17.
There is a famous passage in Plato's Theaetetus in which it is stated
that Teodorus (Plato's teacher) proved the irrationality of
taking all the separate cases up to the root of 17 square feet ... .

14. The Pythagoreans' proof
A more general proof now exists that: The mth root of an integer N is
irrational, unless N is the mth power of an integer n.
That is, it is impossible to express the mth root of an integer N as the
ratio a⁄b of two integers a and b that share no common prime factor except in
cases in which b = 1.

15. Impossible constructions sought by the
ancient Greeks
Three famous questions of Greek geometry were how with compass and
straight-edge
1. to trisect any angle,
2. to construct a cube with a volume twice the volume of a given cube
3. to construct a square equal in area to that of a given circle.
For more than 2,000 years unsuccessful attempts were made to solve these
problems; at last, in the 19th century it was proved that the desired
constructions are logically impossible.

16. Impossible constructions sought by the
ancient Greeks
A fourth problem of the ancient Greeks was to construct an equilateral
polygon with a specified number n of sides, beyond the basic cases n = 3, 4, 5
that they knew how to construct.
All of these are problems in Euclidean construction, and Euclidean
constructions can be done only if they involve only Euclidean numbers.

17. Impossible constructions sought by the
ancient Greeks
Irrational numbers can be Euclidean.
A good example is the irrational number the square root of 2.
It is simply the length of the hypotenuse of a right triangle with legs both one
unit in length, and it can be constructed with straightedge and compass.
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.

18. Impossible constructions sought by the
ancient Greeks
Angle trisection and doubling the cube : Both trisecting the general
angle and doubling the cube require taking cube roots, which are
not constructible numbers by compass and straightedge.
Squaring the circle : Pi is not a Euclidean number. It was proved in 1882 to
be a transcendental number and therefore it is impossible to construct, by
Euclidean methods a length equal to the circumference of a circle of unit
diameter.
Constructing an equilateral n-gon : The Gauss-Wantzel theorem showed in
1837 that constructing an equilateral n-gon is impossible for most values
of n.

19. Euclid's parallel axiom
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".
The question is: can the axiom that two parallel lines "...will not meet even 'at
infinity'" be derived from the other axioms of Euclid's geometry?

20. Euclid's parallel axiom
It was not until work in the nineteenth century by Gauss, Bolyai, Lobachevsky,
and Riemann, that the impossibility of deducing the parallel axiom from the
others was demonstrated.
This outcome was of the greatest intellectual importance.
A proof can be given of the impossibility of proving certain propositions (in
this case, the parallel postulate) within a given system (in this case, Euclid's
first four postulates).

21. Fermat's Last Theorem
Fermat's Last Theorem was conjectured by Pierre de Fermat in the 1600s,
states the impossibility of finding solutions in positive integers for the
equation
Fermat himself gave a proof for the n = 4 case using his technique of infinite
descent, and other special cases were subsequently proved, but the general
case was not proved until 1994 by Andrew Wiles.

22. Richard's paradox
Consider all decimals that can be defined by means of a finite number of
words. Let E be the class of such decimals.
Then E has an infinite number of terms; hence its members can be ordered as
the 1st, 2nd, 3rd, ...
Let X be a number defined as follows : If the n-th figure in the n-th decimal is
p, let the n-th figure in X be p + 1 (or 0, if p = 9).

23. Richard's paradox
Then X is different from all the members of E, since, whatever finite value n
may have, the n-th figure in X is different from the n-th figure in the n-th of
the decimals composing E, and therefore X is different from the n-th decimal.
Nevertheless we have defined X in a finite number of words and therefore X
ought to be a member of E.
Thus X both is and is not a member of E.
— Principia Mathematica, 2nd edition 1927, p. 61

24. Gödel's proof
To quote Nagel and Newman, Gödel's paper is difficult.
Forty-six preliminary definitions, together with several important preliminary
theorems, must be mastered before the main results are reached.
In fact, Nagel and Newman required a 67-page introduction to their
exposition of the proof.
But if the reader feels strong enough to tackle the paper, Martin Davis
observes that "This remarkable paper is not only an intellectual landmark, but
is written with a clarity and vigor that makes it a pleasure to read".

25. Gödel's proof
So what did Gödel prove?
In his own words:
"It is reasonable... to make the conjecture that ...[the] axioms [from Principia
Mathematica and Peano ] are ... sufficient to decide all mathematical
questions which can be formally expressed in the given systems. In what
follows it will be shown that this is not the case, but rather that ... there exist
relatively simple problems of the theory of ordinary whole numbers which
cannot be decided on the basis of the axioms“.

26. Gödel's proof
Gödel compared his proof to "Richard's antinomy“.
An "antinomy" is a contradiction or a paradox; for more see Richard's
paradox.
"The analogy of this result with Richard's antinomy is immediately evident;
there is also a close relationship with the Liar Paradox.
Every epistemological antinomy can be used for a similar proof of
undecidability.

27. Gödel's proof
Thus we have a proposition before us which asserts its own unprovability.
Contrary to appearances, such a proposition is not circular, for, to begin with,
it asserts the unprovability of a quite definite formula.

28. Turing's first proof
The Entscheidungsproblem, the decision problem, was first answered by
Church in April 1935 and preempted Turing by over a year, as Turing's paper
was received for publication in May 1936.
Also received for publication in 1936—in October, later than Turing's—was a
short paper by Emil Post that discussed the reduction of an algorithm to a
simple machine-like "method" very similar to Turing's computing machine
model (see Post–Turing machine for details).
Turing's proof is made difficult by number of definitions required and its
subtle nature.
See Turing machine and Turing's proof for details.

29. Turing's first proof
Turing's first proof (of three) follows the schema of Richard's Paradox.
Turing's computing machine is an algorithm represented by a string of seven
letters in a "computing machine".
Its "computation" is to test all computing machines (including itself) for
"circles", and form a diagonal number from the computations of the non-
circular or "successful" computing machines.
It does this, starting in sequence from 1, by converting the numbers (base 8)
into strings of seven letters to test.

30. Turing's first proof
When it arrives at its own number, it creates its own letter-string.
It decides it is the letter-string of a successful machine, but when it tries to do
this machine's (its own) computation it locks in a circle and can't continue.
Thus we have arrived at Richard's paradox.
If you are bewildered see Turing's proof for more.

31. Before and after Turing's proof
A number of similar undecidability proofs appeared soon before and after
Turing's proof:
April 1935: Proof of Alonzo Church (An Unsolvable Problem of Elementary
Number Theory). His proof was to "...propose a definition of effective
calculability ... and to show, by means of an example, that not every problem
of this class is solvable“.
1946: Post correspondence problem.
April 1947: Proof of Emil Post (Recursive Unsolvability of a Problem of Thue).
This has since become known as "The Word problem of Thue" or "Thue's
Word Problem“. Axel Thue proposed this problem in a paper of 1914.

32. Before and after Turing's proof
Rice's theorem: a generalized formulation of Turing's second theorem.
Greibach's theorem: undecidability in language theory.
Penrose tiling questions
Question of solutions for Diophantine equations and the resultant answer in
the MRDP Theorem.

33. Chaitin's proof
A string is called (algorithmically) random if it cannot be produced from any
shorter computer program.
While most strings are random, no particular one can be proved so, except
for finitely many short ones.
"A paraphrase of Chaitin's result is that there can be no formal proof that a
sufficiently long string is random..." (Beltrami p. 109)

34. Chaitin's proof
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.’
Evidently, the shortest definition of this number must have at least 1000
characters.
However, the sentence within quotation marks, which is itself a definition of
the alleged number is less than 1000 characters in length.

35. Hilbert's tenth problem
The question "Does any arbitrary "Diophantine equation" have an integer
solution?" is undecidable.
That is, it is impossible to answer the question for all cases.
Franzén introduces Hilbert's tenth problem and the MRDP
theorem (Matiyasevich-Robinson-Davis-Putnam theorem) which states that
"no algorithm exists which can decide whether or not a Diophantine equation
has any solution at all".
MRDP uses the undecidability proof of Turing: "... the set of solvable
Diophantine equations is an example of a computably enumerable but not
decidable set, and the set of unsolvable Diophantine equations is not
computably enumerable"

36. In social science
In political science, Arrow's impossibility theorem states that it is impossible
to devise a voting system that satisfies a set of five specific axioms.
This theorem is proved by showing that four of the axioms together imply the
opposite of the fifth.
In economics, Holmström's theorem is an impossibility theorem proving that
no incentive system for a team of agents can satisfy all of three desirable
criteria.

37. In natural science
In natural science, impossibility assertions (like other assertions) come to be
widely accepted as overwhelmingly probable rather than considered proved
to the point of being unchallengeable.
The basis for this strong acceptance is a combination of extensive evidence of
something not occurring, combined with an underlying theory, very
successful in making predictions, whose assumptions lead logically to the
conclusion that something is impossible.
Two examples of widely accepted impossibilities in physics are perpetual
motion machines, which violate the law of conservation of energy, and
exceeding the speed of light, which violates the implications of special
relativity.

38. In natural science
Another is the uncertainty principle of quantum mechanics, which asserts the
impossibility of simultaneously knowing both the position and the
momentum of a particle.
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 science can never be absolutely proved, it
could be refuted by the observation of a single counterexample.
Such a counterexample would require that the assumptions underlying the
theory that implied the impossibility be re-examined.

39. See also
List of unsolved problems in mathematics – Solutions of these problems are
still searched for. In contrast, the above problems are known to have no
solution.
No-go theorem, the corresponding physical notion.

40. References
Proof of impossibility - Wikipedia

41. THANK YOU