Adverb
Too many to be counted (either by reason of being infinite or for practical constraints).
The stars in the sky are uncountably many. Even a lifetime would not suffice to number them all.
(grammar) In an uncountable fashion.
Some nouns can be used both countably and uncountably.
(mathematics) In a way that is incapable of being put into one-to-one correspondence with the natural numbers or any subset thereof.
Source: en.wiktionary.orgA theory that is both -categorical and uncountably categorical is called totally categorical. Source: Internet
A Whiteheadian process is most importantly characterized by extension in space-time, marked by a continuum of uncountably many points in a Minkowski or a Riemannian space-time. Source: Internet
Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. Source: Internet
However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. Source: Internet
In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. Source: Internet
Extension to arbitrary sets of formulas When there is an uncountably infinite collection of formulas, the Axiom of Choice (or at least some weak form of it) is needed. Source: Internet