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Understanding the concept of a 'decidable subset' is essential in the context of computational theory. A subset is considered to be decidable if there exists a Turing machine that can accept all strings in the subset and reject all strings not in it, halting in every case. This characteristic of halting and providing a definite 'yes' or 'no' answer is what makes a subset decidable.

In the context of the exercise, we find that every infinite Turing-recognizable language contains an infinite decidable subset. Recognizing a language means that a Turing machine exists that will halt and accept any string from the language. However, it doesn't require the machine to halt on strings not in the language—it may loop forever. But a 'decidable subset' takes this a step further by halting and providing a definite answer for every input, not just those in the language.

To construct this infinite decidable subset from an infinite Turing-recognizable language, we utilize a procedure that systematically checks each string in lexicographic order. When we find strings that the Turing machine for the original language accepts, we add them to our subset. Since the original language is infinite and we're systematically checking strings, our new subset is guaranteed to be infinite. More importantly, we can construct a new Turing machine that decides this subset, which secures its status as 'decidable.'

A 'Turing machine' lies at the heart of theoretical computer science and is one of the most powerful computing machines defined. It consists of an infinite tape, a head that can read and write symbols on the tape, and a set of rules that determine its actions based on the current state and the symbol it reads. Turing machines can simulate the logic of any computer algorithm, and they are used to define limits of what can be computed.

For our exercise, we refer to a Turing machine 'M' that recognizes an infinite language 'L'. This machine 'M' is able to review each string and determine if it belongs to language 'L' by halting and accepting it. To tackle the idea of a 'decidable subset', we consider a second Turing machine 'M'' that is designed to halt and accept strings from the decidable subset 'S' and reject all others. Through these machines, we bridge the abstract concept of recognition and decidability and show that that a decidable subset can exist within an infinite Turing-recognizable language.

Lexicographic order is an essential concept when discussing languages and sets in computer science. It is a way of ordering sequences of ordered elements, such as strings of characters based on alphabetical order. Lexicographically, 'apple' comes before 'banana', much as in the dictionary. This ordering is applied not only in alphanumeric characters but also in numerical and other systematic strings of elements.

In the exercise, we leverage the lexicographic order to systematically assess the strings within the infinite Turing-recognizable language 'L'. By considering strings in this sequenced manner, we can create a well-defined process to identify our decidable subset 'S'. It ensures that we do not miss any strings that could be part of 'S' and that the subset is indeed infinite, as we're working from an infinite set. The enumeration in lexicographic order is the backbone of creating our subset 'S' in a methodical and organized fashion. It lays the path to proving that even within an unbounded, complex set, order and decisiveness can be found.