[FOM] MathOverflow question on f.o.m.

[Timothy Y. Chow]

On MathOverflow last month, someone asked the following question:

"What is some current research going on in the foundations of mathematics 
about?  Are the foundations of mathematics still a research area, or is 
everything solved? When I think about foundations I'm thinking of reducing 
everything to ZFC set theory + first order logic. Is this still 
contemporary?"

http://mathoverflow.net/questions/254669/what-is-some-current-research-going-on-in-foundations-about

The question was closed as being "too broad" but then was re-opened. 
There are currently two answers, one by Andrej Bauer and one by myself, 
but additional answers would be welcome.

Tim