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A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n.
By extending Shelah's work, Bradd Hart, Ehud Hrushovski and Michael C. Laskowski gave the following complete solution to the spectrum problem for countable theories in uncountable cardinalities. If T is a countable complete theory, then the number I( T , ℵ α ) of isomorphism classes of models is given for ordinals α>0 by the minimum of 2 ...
P satisfies the countable chain condition if every antichain in P is at most countable. This implies that V and V [ G ] have the same cardinals (and the same cofinalities). A subset D of P is called dense if for every p ∈ P there is some q ∈ D with q ≤ p .
List of set identities and relations – Equalities for combinations of sets; Logical conjunction – Logical connective AND; MinHash – Data mining technique; Naive set theory – Informal set theories; Symmetric difference – Elements in exactly one of two sets; Union – Set of elements in any of some sets
If a first-order theory T in a finite or countable signature is κ-categorical for some uncountable cardinal κ, then T is κ-categorical for all uncountable cardinals κ. Morley's proof revealed deep connections between uncountable categoricity and the internal structure of the models, which became the starting point of classification theory ...
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 ...
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]
Being countable implies being subcountable. In the appropriate context with Markov's principle , the converse is equivalent to the law of excluded middle , i.e. that for all proposition ϕ {\displaystyle \phi } holds ϕ ∨ ¬ ϕ {\displaystyle \phi \lor \neg \phi } .