Argument of the diagonal of Cantor

The argument of the diagonal , or diagonal argument was discovered and published by the German mathematician Georg Cantor (1845 - 1918) in 1891. It made it possible this last to give a second Démonstration nondenombrability of the whole of the real numbers, much simpler, according to Cantor itself, than the first which it had published in 1874, and which used arguments of analysis, in particular the Théorème of the encased segments. The diagonal argument was exploited within a more general framework by Cantor in the same article for its theorem on the cardinality of the whole of the parts of a unit.

The diagonal argument applies to a relation or a function (possibly partial) with two arguments to the same field E , or, which returns to same, to a function which with each element of E associates a function defined on E . It uses in an essential way the diagonal of E × E : the whole of the couples ( X , X ) for X in E , from where name.

It was adapted for many demonstrations. Paradoxes which played a part in the foundation of the Set theory as the Paradoxe of Russell (inspired of the Théorème of Cantor) but also the Paradoxe of Richard support on the diagonal reasoning. The Théorème of incomplétude of Gödel uses it for an essential lemma. The Theory of the calculability in fact great use. The diagonal Argument thus became traditional of the demonstration in Mathématiques.

The countable one and the continuous one

One can rest on the decimal Développement realities. Starting from an enumeration of distinct realities, the extraction of the nth decimal of nth reality makes it possible to build a new reality which was not in the enumeration. The decimals can be presented in the form of a semi-infinite table to two entries whose nth line includes/understands the list of the decimals of nth reality. The list of the extracted decimals is read on the diagonal, from where name argument of the diagonal .

A decimal development of a reality is a succession of figures. The argument is in fact valid for an enumeration of continuations of entireties. It is slightly simpler besides in this last case, since the problem of the double representation of decimal does not arise.

Nondenombrability of realities

To show that $\ mathbb {R}$ is noncountable, it is enough to show to it not denombrability of the subset to $\ mathbb {R}$ , therefore to build, for very countable part D of, an element of not belonging to D .

That is to say thus a countable part of enumerated using a continuation $r= (r_i) = \ {r_ 1,r _2,…, r_i,… \}$ . Each term of this following a decimal writing with an infinity of figures after the comma (possibly an infinity of 0 for a decimal number), is:

R _I = 0, R _{I 1} R _{I 2}… R _in…

One now builds a real number X in by considering the n^ième figure after the comma of R _n. For example, for the continuation R :

R ₁ = 0, 0 1 0 5 1 1 0…

R ₂ = 0, 4 1 3 2 0 4 3…

R ₃ = 0, 8 2 4 5 0 2 6…

R ₄ = 0, 2 3 3 0 1 2 6…

R ₅ = 0, 4 1 0 7 2 4 6…

R ₆ = 0, 9 9 3 7 8 1 8…

R ₇ = 0, 0 1 0 5 1 3 0 …

…

The real number X is built by the data of its decimals according to the rule: if decimal N ^E of R _N is different from 1, then decimal N ^E of X is 1, if not N ^E is 2. For example with the continuation above, the rule gives X = 0, 1 2 1 1 1 2 1…

The number X is clearly in interval 1 but cannot be in the continuation { R ₁, R ₂, R ₃,…}, because it is not equal to any the numbers of the continuation: it cannot be equal to R ₁ because the first decimal of X is different from that of R ₁, in the same way for R ₂ by considering the second decimal, etc

Nonthe unicity of the decimal writing for the decimal nonnull (two writings are possible for these numbers, one with all the decimals being worth 0 except a finished number, the other with all the decimals being worth 9 except a finished number) is not a shelf with the preceding reasoning because the diagonal number X cannot be decimal, since its decimal writing is infinite and comprises only figures 1 and 2.

The proof of Cantor

In the article of 1891, where it introduces this reasoning, Georg Cantor builds starting from any enumeration of continuation S of two distinct natures m and W a new continuation, which comprises also only m and W and which was not already enumerated. The reasoning is exactly that described above for realities, simplified owing to the fact that one does not have that two digits - one can take 1 and 2 for m and W - and that this time, like one directly treats continuations, it does not have there more problem of double representation.

The proof spreads in an obvious way to the case of the continuations of elements of a whole to more than two elements (finished or infinite).

One from of thus deduced that the whole of the continuations infinite of 0 and 1 is not countable. However this one corresponds to the binary writing of realities in. however the binary writing of the numbers dyadic S is not single, and if one wants to adapt the reasoning to realities, nothing ensures that built diagonal reality is not dyadic: its binary development could end very well by an infinity of 0 or an infinity of 1.

Cantor does not detail the argument, but it knows in addition that the whole of dyadic realities is countable, and that the meeting of two countable units is countable. It can thus deduce some (in a way more indirect than in the reasoning indicated above) than the unit from realities between 0 and 1 is not countable.

The theorem of Cantor

See also: Theorem of Cantor

Cantor used the argument of the diagonal to show that for all Ensemble S (finished or infinite), all the parts of S , generally noted P (S) , is “strictly larger” than S itself. In other terms, it cannot exist of Surjection of S towards P (S) , and thus not either of injection of P (S) in S . This result is known today under the name of Théorème of Cantor.

For that, it is enough for us to show that, for any function F of S in P (S) , one can build a unit which is not as a whole image of F . Indeed, that is to say the unit has elements X of S such as X does not belong to F (X) . If there was an element has S such as F ( has ) = has , One would lead to a contradiction as well if has belongs to has , that in the contrary case. The unit has thus does not belong to the image of F : this one cannot be surjective.

Here a “picturesque” version more of this argument, if S is the whole of the natural entireties:

That is to say a book comprising as many pages as one wants. Each page is numbered, and, on each one of them, one writes a whole of entireties (all different), of such kind never not to write the same unit twice.
One says that a number NR is ordinary if the unit written in the page NR does not contain NR; in the contrary case, one says that NR is extraordinary . Let us suppose that one wrote on this book all the possible units. The question is: to which category does belong the entirety on the page of which one wrote the whole of the ordinary numbers?

In the countable case, this last form of the diagonal argument is identical to that of the preceding paragraph, where it was shown that the whole of the continuations of two elements is not countable: it is enough to choose an element for “belongs”, the other for “does not belong”. The diagonal argument used for the theorem of Cantor thus differs from that for the continuations only because it is used for a unit S unspecified, instead of the whole of the natural entireties. Is adapted it besides directly to show that the whole of the functions of S in an unspecified whole with more than two elements does not have even cardinality that S .

Calculability

The diagonal reasoning is constructive (one says also effective ). They is completely clearly in the case of the continuations, if each continuation of the enumeration is generated by a calculative process, one has a process to calculate the diagonal continuation. That means that one can theoretically calculate terms of the continuation as many than one wishes, the only limits are material, time and computing power. The diagonal reasoning given for the real ones remains well also constructive. If a succession of realities between 0 and 1 is given to us indeed, with the direction where one can calculate, being given an entirety I , the decimal development of R _I has an arbitrary precision given, then one can calculate a reality not belonging to this continuation (and even a countable infinity of distinct realities all, by deferring to each stage diagonal reality top of the list, which shifts the diagonals). One easily adapts it for a countable succession of realities in general: for example one can use that the diagonal reality of the demonstration above, whose decimal development uses only 1 and 2, thus belongs to] 0,1 and one passes from 0,1 to ''' R ''' by the tangent function, while composing with a function closely connected; however these functions and their reciprocal are calculable, with the direction where one can calculate an arbitrary approximation of the image, being given a sufficient approximation of the antecedent (which depends well-sure on the approximation of the sought image).

The effective character of the diagonal reasoning in did one of the bases of the theory of the Calculabilité, as much for the results of non-existence which are the algorithmic demonstration of indecidability, to start with the proof of the indecidability of the Problème of the stop, that for results of existence, like the theorems of point fixes Kleene.

In a form very close to these theorems of fixed point, it is also an essential argument of the first Théorème of incomplétude of Gödel (which is a logical result of indecidability): in fact the lemma which makes it possible to show the existence of a proposal which involves its own non-prouvability, and which is often called besides lemma of diagonalisation.

Assumption of the continuous one

The demonstration above shows that the whole of the real numbers is “strictly larger” than the whole of the integers. One can put the question to know if there exists a unit whose cardinal is strictly larger than that of

\ mathbb {NR}

and strictly smaller than that of

\ mathbb {R}

. The assumption that there is not, which had with Cantor, is called Hypothèse of continuous the.

In the same way the question of knowing if there exists a whole of cardinality lain strictly between card ( S ) and card ( P ( S )), for a unit S infinite unspecified, led to the assumption of continuous generalized the.

See too

References

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