Cantor's diagonal argument

For an elaboration of this result see Cantor's diagonal argument.

Cantor's diagonal argument shows that the set of such numbers is uncountable.

And yet Cantor's diagonal argument shows that real numbers have higher cardinality.

The paradox can be interpreted as an application of Cantor's diagonal argument.

(A different proof of is given in Cantor's diagonal argument.

Péter exhibited another example (1935) that employed Cantor's diagonal argument.

This appeals strictly to set theoretical notions, and is thus not exactly the same as Cantor's diagonal argument.

This proof is analogous to Cantor's diagonal argument.

This is done using a technique called "diagonalization" (so-called because of its origins as Cantor's diagonal argument).

The lemma is called "diagonal" because it bears some resemblance to Cantor's diagonal argument.

"Cantor's diagonal argument" was the earliest.

See Cantor's first uncountability proof and Cantor's diagonal argument.

The real numbers have no countable enumeration as proved by Cantor's diagonal argument and Cantor's first uncountability proof.

However the set of primitive recursive functions does not include every possible total computable function - this can be seen with a variant of Cantor's diagonal argument.

The original statement of the paradox, due to Richard (1905), has a relation to Cantor's diagonal argument on the uncountability of the set of real numbers.

Adleman's proof is perhaps the deepest result in malware computability theory to date and it relies on Cantor's diagonal argument as well as the halting problem.

Thus Cantor's diagonal argument cannot be used to produce uncountably many computable reals; at best, the reals formed from this method will be uncomputable.

By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable.

Currently there are no natural examples of computable numbers that have been proved not to be periods, though it is easy to construct artificial examples using Cantor's diagonal argument.

This category also extends to attempts to disprove accepted (and proven) mathematical theorems such as Cantor's diagonal argument and Gödel's incompleteness theorem.

Cantor's diagonal argument shows that is strictly greater than , but it does not specify whether it is the least cardinal greater than (that is, ).

But Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable.

Cantor later formulated his second uncountability proof in 1877, known as 'Cantor's diagonal argument', which proved the same thing but employed a method generally regarded as simpler and more elegant than the first.

Given a guaranteed halting language, the computable function which is produced by Cantor's diagonal argument on all computable functions in that language is not computable in that language.

Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers; the second of these is his Cantor's diagonal argument.