## #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.