Constructing a bijection between T and R is slightly more complicated. Instead of mapping The functions f b t are injections, except for f 2 t. This function will be modified to produce a bijection between T and R.
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Since we can construct such a z for any pairing, we know that every pairing has at least one number not in it. Diagonalizations Everywhere It is not clear that I care how many real numbers there are. However, Cantor diagonalization can be used to show all kinds of other things.
For example, given the Church-Turing thesis there are the same number of things that can be done as there are integers. More simply, there are more problems than there are solutions. Diagonalization is so common there are special terms for it. A list that can be shown to be larger than the list of integers is called uncountably infinite, while lists that are the same size as the integers are countably infinite.
Is diagonalization wrong? It feels like sleight of hand, some kind of trick. Let me try to outline some of the ways it could be a trick. Numbers from mathematics have symbolic definitions. Either way, every real number I can ever encounter can be expressed finitely, either by a finite description of defining equations or a finite precision real-world measurement. And if they have finite expressions, then they are countable. In the end, whether you accept diagonalization or not is up to you.
The majority of theoreticians in the world seem to accept it; indeed, not accepting it can earn a bit of ridicule. Accept it only if it convinces you.
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Cantor Diagonal Method
Cantor's diagonal argument