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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 ...
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Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is uncountably , rather than countably , infinite. [ 1 ]
In mathematical logic and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem theorem ; Thoralf Skolem was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the ...
In set theory, Jensen's covering theorem states that if 0 # does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in (Devlin & Jensen 1975).
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1 uncountable set. 2 comments. 2 countable set. 2 comments. 3 The quadratic formula to solve x. 2 comments. 4 equivalent class. 1 comment. 5 solve equations ...
In this constructive set theory with classically uncountable function spaces, it is indeed consistent to assert the Subcountability Axiom, saying that every set is subcountable. As discussed, the resulting theory is in contradiction to the axiom of power set and with the law of excluded middle .