The difference between Countably infinite and Uncountable

When used as adjectives, countably infinite means that is both countable and infinite, whereas uncountable means so many as to be incapable of being counted.

Uncountable is also noun with the meaning: an uncountable noun.

check bellow for the other definitions of Countably infinite and Uncountable

  1. Countably infinite as an adjective (set theory, of a set):

    That is both countable and infinite; having the same cardinality as the set of natural numbers; formally, such that a bijection exists from ℕ to the set.

  1. Uncountable as an adjective:

    So many as to be incapable of being counted.

    Examples:

    "The reasons for our failure were as uncountable as the grains of sand on a beach."

  2. Uncountable as an adjective (mathematics):

    Incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof.

    Examples:

    "Cantor’s “diagonal proof” shows that the set of real numbers is uncountable."

  3. Uncountable as an adjective (grammar, of a noun):

    Describes a meaning of a noun that cannot be used freely with numbers or the indefinite article, and which therefore usually takes no plural form. Example: information.

    Examples:

    "Many languages do not distinguish countable nouns from uncountable nouns."

    "One meaning in law of the supposedly uncountable noun "information" is used in the plural and is countable."

  1. Uncountable as a noun (linguistics):

    An uncountable noun.

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