#006: Is the continuum hypothesis relevant in physics?
Category: Mathematical foundations - Tags: Set theory, Topology
Determine whether the continuum hypothesis can be relevant for physical theories.
Mathematical problem. As part of the general theory, we have determined that any space of experimentally distinguishable cases forms a second-countable \(T_0\) topological space. Therefore we can ask mathematically whether there exists such a topological space where the cardinality of the set is more than countable but less of that of the continuum.
Physical significance. As part of our general theory, we have shown that the cardinality of a set of experimentally distinguishable cases is at most that of the continuum as it must allow a topology that is \(T_0\) and second-countable. The standard formulation of set theory (ZFC) does not tell us whether sets of cardinality between countable and continuum exist, which is then a separate axiom. This extra choice is called the “continuum hypothesis” (CH).
Given this mathematical premise, we should consider whether this choice is relevant for physics. For example, it may be that the set of all possible space-time events (i.e. the structure of space-time) has cardinality that is in between the countable and the continuum. The first step would be to understand whether such a set could be given a suitable topology such that each open set corresponds to an experimentally verifiable statement, as it is done in our general theory.