May two Cohen reals collapse cardinals?

Question

My question is the following:

Let $M$ be a c.t.m. of $\mathsf{ZFC}$. Are there two reals $r_0,r_1 \in \mathbb{R}$ such that $r_i$ is Cohen over $M$ for $i=0,1$ and such that $\omega_1^M$ is countable in $M[r_0,r_1]$?

Thanks

Answer 1

The answer is yes. This is a consequence of the non-amalgamation phenomenon.

Let $M$ be a countable transitive model. Fix an $M$-generic real $z$ collapsing $\omega_1^M$, and let us define two reals $c$ and $d$, so that each is an $M$-generic Cohen real, but together they code $z$.

Enumerate the dense sets $D_0$, $D_1$, and so on for Cohen forcing in $M$, forcing with finite binary sequences under end-extension. Let $c_0$ be the shortest condition in $D_0$. Let $c_1$ be all zeros up to this length, then a $1$, then the first digit of $z$, then extended so as to be in $D_0$. Now extend $c_0$ with all $0$s, up to this length, then a $1$, then the next digit of $z$, and so forth.

Individually, they are $M$-generic Cohen reals, but they are not mutually generic for this forcing, since if we have them together we can observe the coding blocks and get $z$. So $M[c,d]$ collapses $\omega_1^M$, as requested.

I wrote a short article on this kind of argument, which is available at:

• Joel David Hamkins, Upward closure and amalgamation in the generic multiverse of a countable model of set theory, arxiv:1511.01074, 2015.

I had first heard this kind of construction from W. Hugh Woodin, when I was a graduate student, but I'm not sure of the exact provenance. The ideas in that paper led eventually to the following paper:

• Habič, Miha E.; Hamkins, Joel David; Klausner, Lukas Daniel; Verner, Jonathan; Williams, Kameryn J., Set-theoretic blockchains, Arch. Math. Logic 58, No. 7-8, 965-997 (2019). ZBL1468.03063.

Answer 2

Here's a slightly different example. This example has the additional benefit that the analogous argument using measure shows that two random reals can also collapse cardinals.

By Solovay's characterization we know that the set of Cohen reals over a countable transitive model is comeager, and we also know that Baire category notions are preserved by translation (understood here as pointwise addition mod $2$).

Now let $M$ be a countable transitive model and let $z$ be a real that collapses $\omega_1^M$. The sets $\{x\in 2^\omega\mid x \text{ is Cohen over } M\}$ and $\{x\in 2^\omega\mid x\oplus z \text{ is Cohen over } M\}$ are both comeager, where $\oplus$ denotes pointwise addition mod $2$. Hence there must be some real $x$ such that both $x$ and $x\oplus z$ are Cohen over $M$ (just take intersection of the two sets). But any model containing both $x$ and $x\oplus z$ will also contain $z$, so $\omega_1^M$ is countable in $M[x, x\oplus z]$.

Answer 3

Indeed one can prove a more general theorem, from which one can conclude the required question.

Theorem. Suppose $R$ is a real in $V.$ Then there are two reals $a$ and $b$ such that $a$ and $b$ are Cohen generic over $V$, the models $V, V[a], V[b]$ and $V[a, b]$ have the same cardinals and $R \in L[a, b]$.

For a proof see Killing the GCH everywhere with a single real , Theoem 4.1.

To see the theorem implies your requested result, let $V=L[G]$, be the generic extension of $L$ by collapsing some cardinal $\lambda$ into $\omega$ and let $R \in L[G]$ be a real coding such a collapse. Now apply the theorem.