Has there been any research done on the related rates of forcing?

If I force to increase the size of the continuum $\mathfrak{c}$ by 5 $\aleph$'s, say from $\aleph_2$ to $\aleph_7$, how fast does the size of the notion of forcing $\mathbb{P}$ change from the ground model to the forcing extension?

It seems they must just have a constant ratio, but perhaps there are small forcings which have a very big effect.


I don't know if the following results are related, but they might be interesting:

Gitik an I have results, which simply say that (sometimes under the assumption of the existence of large cardinals) one can have a pair (W, V) of models of ZFC, such that adding an $Add(\omega, \kappa)$-generic over V, adds an $Add(\omega, \lambda)$-generic over W, for some $\lambda > \kappa.$

Also, there are results by Shelah and Woodin that one can have a pair (W, V) of models of ZFC, such that W satisfies GCH, V=W[R], for some real R, and In V, the continuum is arbitrary large.

  • $\begingroup$ Thank you for your answer. You say that the rates of change can vary wildly. $\endgroup$ – Erin Carmody Oct 18 '19 at 0:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.