Georg Cantor (note the spelling) was a Russian-born German mathematician who lived from 1845 to 1918. He is remembered for a great many innovations, including the definition of cardinal and ordinal numbers and the creation of set theory.
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One way is Cantor's diagonal argument. See link.
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 study of a number is numeral
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The study of numbers is known as Numerology.