Constructing the iterative hierarchy and zero-sharp

Question

I guess this sort of “definability does not imply existence” (such as zero-sharp) thing is not uncommon? A simple example is the cardinal c that is immediately after aleph-null but below the continuum. Definable but does not exist in universes that satisfy CH (I wrote universes rather than models with a clear nod to JDH). This leads me to a question. Penelope Maddy in her “Believing the Axioms I” (pp 500) writes about constructing the iterative hierarchy from the ground up. By stage omega+ 2, we have the set of reals and we have a well-ordering of type aleph-one. The question is whether or not a one-to-one correspondence between them is included at the next stage, since it is consistent to do so (my paraphrase). If we do, we get CH, else not (so my cardinal c exists). So, in a sense, that’s the stage where we take a call on CH. In the same way, is it possible to pin down the stage where one has to take a call on whether zero-sharp exists?

Answer 1

The $V$-hierarchy is quite coarse: very early on (e.g. at level $\omega+k$ for very small $k$) we can see the answers to such questions.

For getting optimal results along these lines, one very useful fact is the existence of flat pairing functions: these are pairing functions $\langle\cdot,\cdot\rangle$ such that whenever $\alpha$ is infinite and $x,y\in V_\alpha$ then $\langle x,y\rangle\in V_\alpha$. Using a flat pairing function we get for example that every binary relation on $V_\alpha$ for $\alpha$ infinite is an element of $V_{\alpha+1}$ (which is clearly the best we could hope for).

This lets us argue as follows:

• There is a canonical partition $P$ of $V_{\omega+1}$ into $\omega_1$-many pieces. (That is, $P$ is definable and ZFC proves that it has the above property.)

• CH is equivalent to the statement "There is a relation $R\subseteq (V_{\omega+1})^2$ such that $(i)$ for each $x\in V_{\omega+1}$ the set $\{y: R(x,y)\}$ is a $P$-class and $(ii)$ if $x_1\not=x_2$ and $R(x_1,y_1)$ and $R(x_2,y_2)$ hold then $y_1$ and $y_2$ lie in different $P$-classes.

• Such $P$ and $R$ - if they exist - live in $V_{\omega+2}$, and moreover their behavior is "verifiable in $V_{\omega+2}$."

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Precisely, we've shown:

If $M,N\models ZFC$ with $(V_{\omega+2})^M=(V_{\omega+2})^N$ then we have $M\models CH$ iff $N\models CH$.

(Note that when I say "$(V_{\omega+2})^M=(V_{\omega+2})^N$" I'm requiring agreement of the elementhood relation as well.)

Conversely, this is optimal:

There are models of ZFC with the same $V_{\omega+1}$ which differ about CH.

Namely, if $M\models \neg CH$ then there is a forcing extension $N$ of $M$ with the same reals which satisfies $CH$.

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What about $0^\sharp$?

This turns out to be easier: one of the definitions of $0^\sharp$ is projective (indeed, $\Delta^1_3$ - meanwhile $\{0^\sharp\}$ is a $\Pi^1_2$ singleton), and so we know whether it exists as soon as we know what reals we have. Thus, "$0^\sharp$ exists" is decided at level $\omega+1$:

If $M,N\models ZFC$ with $(V_{\omega+1})^M=(V_{\omega+1})^N$, then $M$ and $N$ agree on whether $0^\sharp$ exists.

Note that this demonstrates that there's really no connection between the level at which a principle is decided and the consistency strength of that principle.

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OK, what if we replace the $V$-hierarchy with something finer?

One issue here is that finer hierarchies don't (generally) provably exhaust the universe. The [$L$-hierarchy is pretty fine (and we can go even finer in various ways), but $V=L$ (or even $(V_{\omega+1})^L=V_{\omega+1}$) is undecidable in ZFC. So the finer approach only works if we add additional "structurally limiting" axioms to ZFC.

But those axioms tend to be so strong that they decide outright the principles we're interested in.

This leads us unfortunately to:

As far as I can tell, there's no natural version of this question which yields "fine distinctions:" the $V$-hierarchy makes everything happen "as soon as possible," while finer hierarchies tend to require structural limitations which seem to decide all natural principles outright.

EDIT: As Andres says below, that's hiding quite a lot to the point of being at least ethically incorrect. I'll rewrite that part when I have a chance, but the relevant material is sufficiently outside my comfort zone that I can't do that immediately.