"A collection of abstract symbols, together with a set of rules for assembling the symbols into strings" (J. CASTI, 1994, p.121).
CASTI adds to this definition: "Such a system has four components:
"Alphabet: This is simply the set of abstract symbols themselves. These may be as primitive as the symbols + and –. But they could also be much more concrete, such as the characters of the latin alphabet, along with the symbols for punctuation, logical combination and the like
"Grammar: The grammatical rules of the system specify the valid ways in which we can combine the symbols of the alphabet to form finite strings (words), as well as how such strings may be combined into large statements. Statements of this type are termed well/formed.
"Axioms: A set of well-formed statements that are taken as given strings of the system are called axioms… Thus, an axiom is a string of symbols that we accept as a valid statement without its having to be proved (italics ours).
"Rules of inference: These are the procedures by which we combine and change axioms and other well-formed statements into new well-formed strings" (1994, p.121).
CASTI adds an important caveat: "… it is evident that in a very definite sense the theorems of a formal system are already present in the axioms and rules of inference. All a proof sequence does it to make explicit that which is already implicit" (p.123).
Thus formal systems are closed and, for this reason, submitted to the limits of GÖDEL's incompleteness theorem.
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Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]
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