Why is the set of infinite length strings over zeros and ones considered uncountably infinite?
The set of infinite length strings over zeros and ones is considered uncountably infinite due to its cardinality being larger than that of the set of natural numbers. This concept can be understood by examining Cantor's diagonal argument, which demonstrates that there are more real numbers than natural numbers. By extension, the set of infinite length strings over zeros and ones can also be shown to have a larger cardinality than the set of natural numbers.
To explain this concept further, let us consider a hypothetical scenario where we attempt to list all the infinite length strings over zeros and ones. We can start by listing the strings in lexicographic order, where the length of the string is increasing. For example, we can begin with the empty string, followed by all length 1 strings (0, 1), then all length 2 strings (00, 01, 10, 11), and so on.
Now, let us construct a string that is not in our list. We can do this by considering the diagonal elements of our list. The diagonal element of the first row will be different from the first element of our first string, the diagonal element of the second row will be different from the second element of our second string, and so on. By flipping the bits of the diagonal elements, we can construct a new string that is not present in our original list.
This new string is distinct from all the strings in our list, as it differs from each string in at least one position. Therefore, our original attempt to list all the infinite length strings over zeros and ones is incomplete, as we have missed at least one string. This demonstrates that the set of infinite length strings over zeros and ones is uncountably infinite.
To further illustrate this concept, consider the binary representation of real numbers between 0 and 1. Each real number in this interval can be represented as an infinite length string over zeros and ones, where each digit represents a binary place value. For example, the number 0.5 can be represented as the infinite length string 0.100000… (with an infinite number of zeros after the decimal point).
Since there are uncountably infinite real numbers between 0 and 1, and each real number corresponds to a unique infinite length string over zeros and ones, we can conclude that the set of infinite length strings over zeros and ones is also uncountably infinite.
The set of infinite length strings over zeros and ones is considered uncountably infinite because its cardinality exceeds that of the set of natural numbers. This can be demonstrated through Cantor's diagonal argument and the correspondence between infinite length strings and real numbers between 0 and 1.
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