# 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

# Re: Paper and slides on indefiniteness of CH

On Oct 15, 2014, at 6:56 AM, Sy David Friedman wrote:

Dear Harvey,

Maybe I can be of some help with the set theory question (Hugh, feel free to correct or amplify what I say):

If the GCH holds at some infinite cardinal $\kappa$ in $M$ then one can add a new subset of $\kappa^+$ without adding a new subset of $\kappa$ and without collapsing cardinals. If the GCH holds at unboundedly many cardinals in $M$ then $M$ has a cardinal-preserving extension which is not a set-generic extension.

Maybe you need GCH holds on a club class? Otherwise I do not see that the Easton products do not collapse for example the double successors of limits. But maybe you have something else in mind at limit stages?

So the difficulty is with models $M$ in which the GCH fails at all sufficiently large cardinals.

I agree, these seem like the very difficult cases.

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

I introduced a cardinal-preserving method for forcing clubs through $\omega_2$ with finite conditions, even without CH in the ground model. My motivation was to try to do this for $\text{Ord}$ instead of for just omega_2, preserving the powerset axiom. (I was looking for a new characterisation of $0^\#$.) So I was essentially asking your question back then. Unfortunately there were 2 obstacles: I didn’t even know how to do this for $\omega_3$ or for $\omega_2$ without killing CH (see the last 2 questions in my paper cited above). The good news is that Krueger and Mota recently solved the latter problem; we are currently thinking about $\omega_3$. So my conjecture is: There is a cardinal-preserving class-forcing with finite conditions that does not reduce to a set-forcing and preserves ZFC. I admit that this is very hard, but there is no hint of an obstruction to it. At the same time, I confess that I don’t know how to do it.

I agree that may be how the question go. Actually I think Aspero has the best partial results now, he can prove that for each n there is a cardinal preserving forcing which has a new subset of $\aleph_n$ which is not $\aleph_n$-cc generic over V.

However to generalize the method it looks like one might need square at $\alpha_{\omega}$ etc. So there may be some rather serious obstructions.

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

Dear Harvey,

Maybe I can be of some help with the set theory questions (Hugh, feel free to correct or amplify what I say):

On Wed, 15 Oct 2014, Harvey Friedman wrote:

Hugh wrote:

Suppose $M$ is a ctm and $M \vDash \textsf{ZFC}$. Must $M$ have an outer model of $\textsf{ZFC}$ which is cardinal preserving and not a set forcing extension? Number Theory is full of problems which seem completely out of reach. And one can easily generate such problems in a foundational guise.

I have a couple of questions about Hugh’s question.

1. The question as formulated involves both forcing extensions and general (outer) extensions. Are there appropriate formulations that

a. Do not refer to forcing at all.

Here is one (but it is cheating, and I’ll explain why below): Does every ctm $M$ of ZFC have an outer model $N$ of ZFC with the following covering property? For some cardinal $\kappa$ of $N$, every function $f\in N$ (with ordinal domain) is covered by a multi-valued function $g \in M$ with the same domain and at most $\kappa$ values? (I.e., for each $\alpha$ in the domain of $f$, $g(\alpha)$ has size at most $\kappa$ and $f(\alpha)$ is an element of $g(\alpha)$).

This is “cheating” because I have just used a theorem of Bukovsky to reformulate “set-generic extension” in terms of a covering property!

b. Given any ctm $M$, refer only to forcing extensions.

Here is one: Does every $M$ have a cardinal-preserving forcing extension which adds a new set but no new real? This was first asked by Matt Foreman. I think it is still open.

2. What are the known natural conditions on $M$ that are known to be sufficient? Also for variants in 1 above.

If the GCH holds at some infinite cardinal $\kappa$ in M then one can add a new subset of $\kappa^+$ without adding a new subset of kappa and without collapsing cardinals. If the GCH holds at unboundedly many cardinals in $M$ then $M$ has a cardinal-preserving extension which is not a set-generic extension.

So the difficulty is with models $M$ in which the GCH fails at all sufficiently large cardinals.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

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.

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?

Harvey