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What does countable mean?

Looking for the meaning or definition of the word countable? Here's what it means.

Adjective
  1. Capable of being counted; having a quantity.
  2. (mathematics) Finite or countably infinite; having a one-to-one correspondence (bijection) with a subset of the natural numbers.
  3. (mathematics) Countably infinite; having a bijection with the natural numbers.
  4. (grammar) Freely usable with the indefinite article and with numbers, and therefore having a plural form.
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Examples
Mathematicians divide infinite sets into two categories, countable and uncountable sets.
A full hierarchical perspective suggests a continuum of variation rather than a countable number of objects.
The optimum fibre density should give an average of about one or two countable fibre per fields of view at 5 000 magnification.
The world, Heidegger says, is not the mere collection of the countable or uncountable, familiar and unfamiliar things that are at hand.
As might be expected Cantor went on to find other kinds of infinities that were not countable by the rules he had established.
In order to cast a countable vote, a voter must read and follow a set of rudimentary instructions.

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