In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise.
“How can I stay in any field and continue to prove theorems if the fundamental notions I’m using are problematic?” asks Peter Koellner, a professor of philosophy at Harvard University who specializes in mathematical logic.
To Settle Infinity Dispute, a New Law of Logic is an interesting article in Quanta Magazine exploring the disagreements among mathematicians about the continuum hypothesis.
Who wins in the ever-so-relevant showdown between forcing axioms and the inner-model axiom, "V=ultimate L"?"