Forcing and gaps in $2^\omega$

This is a talk at the CUNY Set Theory Seminar, December 2nd, 2011.
Notes

For $a,b\in 2^\omega$, we say that $a$ is eventually dominated by $b$, denoted by $a\leq^*b$, if $a(n)\leq b(n)$ for all but finitely many $n$. Let $\mathcal A=\langle a_\alpha\mid \alpha<\kappa\rangle$ and $\mathcal B=\langle b_\beta\mid \beta<\lambda\rangle$, where $\kappa$ and $\lambda$ are infinite regular cardinals, be a pair of sequences in $2^\omega$. The pair $(\mathcal A,\mathcal B)$ is called a $(\kappa,\lambda)$-pregap if $a_{\alpha_1}\leq^*a_{\alpha_2}\leq^*b_{\beta_2}\leq^* b_{\beta_1}$ for all $\alpha_1<\alpha_2<\kappa$ and $\beta_1<\beta_2<\lambda$. That is, we have: $$a_0\leq^* a_1 \leq^*\cdots\leq^* a_\alpha \leq^*\cdots \leq^* b_\beta\leq^* \cdots \leq^* b_1 \leq^* b_0$$ We say that a set $c\in 2^\omega$ separates the pregap $(\mathcal A,\mathcal B)$ if $a_\alpha\leq^* c\leq^* b_\beta$ for all $\alpha<\kappa$ and $\beta<\lambda$. That is, we have: $$a_0\leq^* a_1 \leq^*\cdots\leq^* a_\alpha \leq^*\cdots\leq^* c\leq^*\cdots \leq^* b_\beta\leq^* \cdots \leq^* b_1 \leq^* b_0$$ If there is no such set $c$, then we say that the pregap $(\mathcal A,\mathcal B)$ is a $(\kappa,\lambda$)-gap. Much of the literature on gaps also studies gaps in $\omega^\omega$ under the eventual domination ordering. However, it is not difficult to see that for infinite regular cardinals $\kappa$ and $\lambda$, there is a $(\kappa,\lambda)$-gap in $\omega^\omega$ if and only if there is one in $2^\omega$. Thus, we can concentrate on gaps in $2^\omega$ without loss of generality. In what follows we tacitly associate elements of $2^\omega$ with subsets of $\omega$.

Theorem: (Hadamard, 1894) There are no $(\omega,\omega)$-gaps.

Proof: Consider a pregap $(\mathcal A,\mathcal B)$, where $\mathcal A=\langle a_n\mid n<\omega\rangle$ and $\mathcal B=\langle b_m\mid m<\omega\rangle$. Let $\overline b_m$ denote the complement of $b_m$ and define $c_n=a_n\setminus (\bigcup_{m\leq n} \overline b_m)$. It is now easy to see that $c=\bigcup_{n<\omega} c_n$ separates $(\mathcal A,\mathcal B)$. $\square$

strong>Theorem: (Hausdorff, 1909) There is an $(\omega_1,\omega_1)$-gap.

For a proof see [1] (Section 29).

Here, we focus on the interaction between $(\omega_1,\omega_1)$-gaps and forcing. In particular, we are interested here in the following questions:

Question: Can we create a generic $(\omega_1,\omega_1)$-gap by $\omega_1$-preserving forcing?

Let us call an $(\omega_1,\omega_1)$-gap destructible, if there is an $\omega_1$-preserving forcing which adds a set separating it. Note that every $(\omega_1,\omega_1)$-gap is trivially destructible, if we remove the requirement that the forcing is $\omega_1$-preserving, by collapsing $\omega_1$ to $\omega$. We call an $(\omega_1,\omega_1)$-gap indestructible if it is not destructible. Kunen showed (1976) that Hausdorff's gap is indestructible.

Question: Are there destructible $(\omega_1,\omega_1)$-gaps?

Question Can we force to make an $(\omega_1,\omega_1)$-gap indestructible?

The material in this talk draws mainly on Teruyuki Yorioka's thesis [bibcite key=yorioka:thesis] and Marion Scheepers' survey paper [2].

References

1. T. Jech, Set theory. Berlin: Springer-Verlag, 2003, p. xiv+769.
2. M. Scheepers, “Gaps in $ω^ω$,” in Set theory of the reals (Ramat Gan, 1991), vol. 6, Ramat Gan: Bar-Ilan Univ., 1993, pp. 439–561.