Page 39 - GALIET PHYSICS BLOSSOMS I+

A Formal Axiomatic System
The idea of a formal axiomatic approach, what it comprises, and how it is constructed and interpreted is as follows:
A formal axiomatic system comprises:
1. Initial basic statements or assumptions (axioms)
a. Axioms are just a ‘set of symbols,’ which can be strung together in ‘strings’
or phrases according to a set of rules. Strings are just symbols, without
meaning. 2. Rules of argument
a. These rules
i. Transform axioms or ‘set of symbols’ into other prepositions.
ii. Manipulate the symbols to get to other strings, or they derive a large number of other strings from a given string.
3. A Conclusion (Final Statement)
a. The generation of a final string of symbols
Construction and Interpretation of an Axiomatic System
We start with assigning a simple set of symbols or axioms (a sort of alphabet, if you wish) to the topic (situation). This alphabet or set of axioms or symbols can then be constructed in phrases or strings subject to certain rules. There is no meaning to the phrases initially. From these phrases or axioms, made up of symbols and strings of symbols, following certain rules, we can derive more sets of strings and so on. We can think of each string as a node (an axiom) in a network of propositions where the rules generate links to other nodes, and eventually generate a conclusion or a final string of symbols. What matters is how we manipulate the strings of symbols and what the initial symbols are.
How we interpret the system is that in any given system there will be inaccessible strings or strings of symbols that are beyond the system or “off the grid.” Anything “off the grid” is neither True nor False, that is, it is unprovable or meaningless. In other words, with “off-the-grid” axioms or premises, we cannot reach a conclusion, because they are outside the system, or “off the grid.” Briefly, these are “outliers” strings outside the system. What this basically says is that there are legitimate ‘phrases’ in this language that can never be
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