#007: Can the current logic axioms be simplified?
Category: Mathematical foundations - Tags: General theory, Mathematical logic, Boolean algebra, Heyting algebra, sigma-algebra
See whether the axioms of logic can be reduced (and maybe put in a form more “palatable” to logicians?).
Mathematical problem. The current formulation (see the book) defines a context as a set of propositions of potentially any cardinality from which a countably generated subset is extracted. Part of the reason is that it was hard to find mathematicians that were able to tell us precisely what are the conditions for closure. To be safe, we specified closure as one of the axioms.
Since, in the end, the only closure that we will need is under countable operations, it may be possible to bootstrap the system simply with a countably generated Heyting algebra (which represents a closed set of verifiable statements). This requires that any Heyting algebra can be embedded into a unique countably generated \(\sigma\)-algebra. At that point, the context is simply (in our nomenclature) the theoretical domain that contains everything.
Physical significance. Currently, the idea is that we start with a logical context which is the set of all the statements we may want to make within a specific theory. The physically meaningful statements are subsets of a context. For example, the context may contain statements that are not experimentally verifiable (e.g “there exist other universes non-interacting with ours”), but when constructing experimental domains these non-distinguishable possibilities merge together to match the resolution provided by the statements.
The proposal here is to start only with verifiable statements. What needs to be understood is whether the logical relationship defined on the verifiable statements are enough to infer the logical relationship on all (theoretical) statements that can be generated with countably many operations (i.e. statements for which a test in principle can be constructed).
The existence of a unique closure would also guarantee that we are not accidentally ruling out some theories, and that theoretical statements can always be understood to follow the law of excluded middle.