# Re: Paper and slides on indefiniteness of CH

On Oct 14, 2014, at 3:50 PM, Harvey Friedman wrote:

Hugh just wrote:

“I would argue instead that this is simply a sort of coming of age for Set Theory; i.e. we can now pose simple questions about models of Set Theory which seem completely out of reach.

Number Theory is full of such problems. And one can easily generate such problems in a foundational framework”

I and I think many readers of this discussion, would very much like to see such “simple questions about models of Set Theory which seem completely out of reach” explained in generally understandable terms, nicely laid out in one message. I especially would like to see the simplest such question you have in mind.

Suppose $M$ is a ctm and $M \vDash \textsf{ZFC}$. Must $M$ have an outer model of ZFC which is cardinal preserving and not a set forcing extension?

I’m not sure what the meaning is of the second sentence in the second paragraph above. Are you simply referring to the first paragraph above?

Number Theory is full of problems which seem completely out of reach.  And one can easily generate such problems in a foundational guise.

Hugh