One way is Cantor's diagonal argument. See link.
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It is uncountable, because it contains infinite amount of numbers
The countable nouns are nouns with a singularand a plural form.The uncountable nouns are also called mass nouns.
No, the noun 'rain' is a singular, uncountable (mass) noun as a word for water drops falling from clouds; a word for percipitation.The plural noun 'rains' is a plural uncountable (mass) noun as a word specifically for seasons or periods of rain.
No, the set of all irrational numbers is not countable. Countable sets are those that can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). The set of irrational numbers is uncountable because it has a higher cardinality than the set of natural numbers. This was proven by Georg Cantor using his diagonalization argument.
The word butter is an uncountable noun. Thus, it doesn't have a separate plural form.