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<p>without the <a href="page.php?w=axiom_of_choice">axiom of choice</a>, but the equivalence of the third and fourth cannot be proved without additional choice principles.</p>

<p><big>Properties</big></p>
<p>If an uncountable set X is a subset of set Y, then Y is uncountable.</p>

<p><big> Examples </big></p>
<p>The best known example of an uncountable set is the set  of all <a href="page.php?w=real_number">real number</a>s; <a href="page.php?w=Cantor%27s_diagonal_argument">Cantor's diagonal argument</a> shows that this set is uncountable. The diagonalization proof</p><p>
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