Thus, large cardinals can be used to prove that a given sentence \(\varphi\) does not imply another sentence \(\psi\), modulo ZFC, by showing that ZFC plus \(\psi\) implies the consistency of some large cardinal, whereas ZFC plus \(\varphi\) is consistent assuming the existence of a smaller large cardinal, or just assuming the consistency of ZFC. In other words, \(\psi\) has higher consistency strength than \(\varphi\), modulo ZFC. Then, by Gödel’s second incompleteness theorem, ZFC plus \(\varphi\) cannot prove \(\psi\), assuming ZFC plus \(\varphi\) is consistent.