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In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC. Sierpiński showed that they exist if the continuum hypothesis is true. On the other hand, they do not exist if Martin's axiom for ...
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
One of the earliest results in set theory, published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets. [2] An infinite set is called countable if there is a function that gives a one-to-one correspondence between and the natural numbers, and is uncountable if there is no such correspondence function.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers, which Cantor began. Georg Cantor published this proof in 1891, [1] [2]: 20– [3] but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.
The decidable membership of = makes the set also not countable, i.e. uncountable. Beyond these observations, also note that for any non-zero number a {\displaystyle a} , the functions i ↦ f ( i ) ( i ) + a {\displaystyle i\mapsto f(i)(i)+a} in I → N {\displaystyle I\to {\mathbb {N} }} involving the surjection f {\displaystyle f} cannot be ...
It is called countable if the set of function and relation symbols in it is countable, and in general the cardinality of a signature is the cardinality of the set of all the symbols it contains. A first-order theory consists of a fixed signature and a fixed set of sentences (formulas with no free variables) in that signature.
The concept of countability led to countable operations and objects that are used in various areas of mathematics. For example, in 1878, Cantor introduced countable unions of sets. [67] In the 1890s, Émile Borel used countable unions in his theory of measure, and René Baire used countable ordinals to define his classes of functions. [68]