22
votes
4answers
973 views

What is the definition of a large cardinal axiom?

In different books one can find different implicit definitions for a large cardinal axiom. My question is that which one of these definitions are more popular or standard amongst set theorists? Any ...
9
votes
2answers
593 views

Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?

I was recently told that the following (due to M. Viale) is a nice theorem: Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$. Let $G$ be generic for a proper ...
14
votes
2answers
2k views

Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
31
votes
7answers
5k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...