Sentence examples for denumerable from inspiring English sources
adjective
Capable of being assigned numbers from the natural numbers. Especially applied to sets where finite sets and sets that have a one-to-one mapping to the natural numbers are called denumerable.
- The empty set is denumerable because it is finite
Since an ordinary first-order language has a denumerable supply of open formulas, at most denumerably many sets (in any given domain) can be specified in this way.
In 1873 Cantor demonstrated that the rational numbers, though infinite, are countable (or denumerable) because they may be placed in a one-to-one correspondence with the natural numbers (i.e., the integers, as 1, 2, 3,...).
The symbol ℵ0 (aleph-null) is standard for the cardinal number of N (sets of this cardinality are called denumerable), and ℵ (aleph) is sometimes used for that of the set of real numbers.
First Cantor rigorously demonstrated that the set of rational numbers (fractions) is the same size as the counting numbers; hence, they are called countable, or denumerable.
Since an uncountable infinity is much larger than a countable one, it follows that Turing machines (and hence digital computers) can compute only a tiny portion of all functions (over denumerable domains, such as natural numbers or strings of letters).
Blum et al.'s result is equivalent to demonstrating that all functions over denumerable domains — including the uncountably many functions that are not Turing-computable — are computable by Blum et al.'s "computing" systems, which are allowed to manipulate the exact values of arbitrary real numbers.
Aleksandrov had his first major mathematical success in 1915, proving a fundamental theorem in set theory: Every non-denumerable Borel set contains a perfect subset.
Now in ZF one can prove the existence of sets with a non-denumerable number of elements such as the set ℜ of real numbers.
In the same short paper (1892), Cantor presented his famous proof that R is non-denumerable by the method of diagonalisation, a method which he then extended to prove Cantor's Theorem.
"One might say," Wittgenstein says, "I call number-concept X non-denumerable if it has been stipulated that, whatever numbers falling under this concept you arrange in a series, the diagonal number of this series is also to fall under that concept" (RFM II, §10; cf. II, §§30, 31, 13).
Can a going-somewhere be composed by an (even more-than-denumerable) infinity of going-nowheres?
