Applications of advanced set theory to other areas of math

Question

Set theory, from what I understand, aims to provide a solid foundation for mathematics. However, after a certain point, it seems to take on a life of its own. Ideas/objects I would list as illustrations of this are GCH and the existence of strongly inaccessible cardinals. Do these ideas/objects have any consequence for other fields of mathematics? I.e., would the life of a mathematician in any other field be changed if we discovered that, say, strongly inaccessible cardinals do in fact exist?

Answer 1

I tried to find examples that entangles set theory and fields outside of mathematical logic, but it might not exactly go on.

• Various independence results outside set theory could be examples. This includes Whitehead problem and Naimark's problem. See the relevant Wikipedia article for more examples.

• Dehornoy order is an example. Huge cardinal influences its discovery.

• It is known that the product of two CW-complexes is not necessarily a CW-complex. The precise condition when the product of two CW-complexes is again a CW-complex is related to a cardinal characteristic of the continuum (specifically, the bounding number $\mathfrak{b}$.) See this presentation slide for details.

• The rearrangement number, which emerges from a question on Mathoverflow, is also an example.

• There is an application of the continuum hypothesis to a theoretical physics. (I do not know this result is physically meaningful, but it could have philosophical implications.)