elementary set theory - What do finite, infinite, countable, not . . . Clearly every finite set is countable, but also some infinite sets are countable Note that some places define countable as infinite and the above definition In such cases we say that finite sets are "at most countable"
Prove that the set of all algebraic numbers is countable $\begingroup$ You used "a countable union of countable sets is countable" which in its general form requires AC, though that can be dispensed with in this case The proof suggested by the hint has a (somewhat) more constructive character
What does it mean for a set to be countably infinite? A countable set has an injection with the natural number and is therefore in a sense 'of the same size' as the natural numbers The uncountable sets are the sets not having this property and are therefore 'bigger'
How to prove that the set of rational numbers are countable? Any set that can be put in one-to-one correspondence in this way with the natural numbers is called countable In some sense, this means there is a way to label each element of the set with a distinct natural number, and all natural numbers label some element of the set $\endgroup$
Is $\mathbb{Q}^n$ countable? - Mathematics Stack Exchange Is the following a valid countable dense subset of (0,1) the open unit interval? 2 The normed space of convergent sequences of real numbers is separable (proof verification)
Prove that the union of countably many countable sets is countable. So the integers are countable We proved this by finding a map between the integers and the natural numbers So to show that the union of countably many sets is countable, we need to find a similar mapping First, let's unpack "the union of countably many countable sets is countable": "countable sets" pretty simple