A button for individual reals

Question

Hamkins introduced the notion of a "button" in forcing. This is a set-theoretic statement that can be forced, and can never be made false by further forcing. An example is $V \not= L$. Another example (with parameters) is "$S$ is a nonstationary subset of $\kappa$."

Question: Is there an analogue of the second example for reals? I mean a property $\varphi(v,a)$, where $a$ is a parameter, such that there is some forcing $\mathbb P$ and some class of reals $C$ such that for all $r \in C$, $\Vdash_{\mathbb P} \varphi(\check r, \check a)$, and for all $\mathbb P$-names $\dot{\mathbb Q}$ for partial orders, $\Vdash_{\mathbb P * \dot{\mathbb Q}} \varphi(\check r, \check a)$?

Answer 1

The way you've stated it, you haven't said that $\varphi$ isn't already true, and so technically any tautological statement would work. But I assume that you want $\varphi(r,a)$ to start out false in the original model $V$. This would be an unpushed button, which hasn't yet been pushed.

It is consistent with ZFC that there are no such statements (with real parameters). This is exactly equivalent to the axiom $\text{MP}(\mathbb{R})$, which is discussed in my paper

• J. D. Hamkins, A simple maximality principle, J. Symbolic Logic, vol. 68, iss. 2, pp. 527-550, 2003.

Namely, the axiom $\text{MP}(\mathbb{R})$ asserts that all buttons with real parameters have already been pushed.

Meanwhile, it is also consistent that there are many such statements. For example, let $C$ be the set of reals coding ordinals, and let $\varphi(r)$ assert that $\aleph_\alpha^L$ is collapsed, where $r$ is a code for $\alpha$. This is not true in $L$, but it is possibly necessary by forcing. More generally, if $W$ is any definable inner model that is sufficiently absolute, then you can refer to collapsing cardinals of $W$ in a similar way.

You might be interested in having large independent families of buttons, which means that they can each be pushed without pushing any others, as desired, in any forcing extension. These were important in the work I did with Benedikt Löwe on the modal logic of forcing.

For example, if you have a stationary partition $\omega_1=\bigsqcup_\alpha S_\alpha$ of $\omega_1$ in $L$, then you can make the assertion $\varphi(r)$ that $S_\alpha$ is no longer stationary, using the least such partition, where $r$ codes $\alpha$. If $C$ has one code for each ordinal, then these buttons can be controlled independently, since the usual way of killing stationarity will preserve all stationary subsets of the complement.

You can obviously modify this trick in many other ways. For example, add a large family $C$ of mutually generic Cohen reals, and then let $\varphi(r)$ for $r\in C$ assert that the least Suslin tree in $L[r]$ is no longer Suslin.