  Each Turing machine can be uniquely described by its transition table: for each state, \(q\), and each symbol, \(\sigma , \delta(q,\sigma)\) is the new state, the new symbol, and the head displacement. These transition tables, can be written as a finite string of symbols, giving the complete set of instructions of each Turing machine. Furthermore, these strings of symbols can be listed in lexicographic order as follows: \(M_{1}, M_{2}, M_{3},\ldots\), where \(M_{i}\) is the transition table, i.e., the complete set of instructions, for Turing machine number \(i\). The transition table for \(M_{i}\) is the program for Turing machine \(i\), or more simply, the \(i\)th program.