With the help of sets of good indiscernibles above a certain height, we show that Chang conjectures involving four, finitely many, or an ω-sequence of cardinals have a much lower consistency strength with ZF than they do with ZFC. We will prove equiconsistency results for any finitely long Chang conjecture that starts with the successor of a regular cardinal. In particular, any Chang conjecture of the form (κn, . . . , κ0) (λn, . . . , λ0), where κn is the successor of a regular cardinal, in a… CONTINUE READING