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Instructions
Compare the cardinality of integers to the cardinality of real numbers. In mathematics it has been determined that the set of integers is countably infinite while the set of real numbers is not countably infinite. That is, both sets are infinite but the set of integers is countably infinite while it is not possible to count all the numbers in the set of real numbers.
Refer to Cantor's Diagonalization Argument to understand the difference between the countability of the set of integers and the set of real numbers. Cantor based his argument on first visualizing numbers written out in a grid. Rather than counting all the numbers, the numbers along each diagonal were counted. In doing so Cantor was able to show that some sets are more infinite than others, meaning that some infinite sets have a higher cardinality than others. In this case, the set of real numbers has a higher cardinality than the set of integers. In fact the set of real numbers between 0 and 1 has a higher cardinality than the entire set of integers.
Write the cardinality of all natural numbers as aleph null -- that is, write the aleph, the first letter of the Hebrew alphabet, with a subset of 0. This symbol is also called aleph nought. Just as we use the infinity symbol to denote infinity, aleph null is used to represent the infinitely high number that is the cardinality of all natural numbers.
Write the cardinality of the set of real numbers as a lowercase c. Since we already know there is not a 1-to-1 correspondence with aleph null -- the infinite number that represents all the integers -- we know that the set of real numbers cannot be aleph null. Technically, this number is aleph one, written as an aleph with a subset of one. For simplicity's sake, this is represented by the lowercase letter c. Just like with aleph null and the infinity symbol, this symbol stands for an infinitely large number.