Forcing is a mechanism of obtaining independent results over the commonly accepted foundation of mathematics, the Zermelo-Fraenkel axiom system with the axiom of choice. Since its formulation by Cohen [1], forcing has become a powerful tool in advanced set theory as well as other related fields and proved numerous results.

Roughly speaking, forcing usually works in the following way: to prove that a statement $\neg \phi$ is unprovable from ZFC, one builds a model of ZFC together with $\phi$. One starts with a ground model $\mathbb{V}$ and force with a generic filter $G$ to get a bigger model, the generic extension $\mathbb{V}[G]$, which witnesses the failure of $\neg \phi$. The generic filter $G$ usually collects information which approximates the desired statement. This process is in a sense similar to a field extension. For instance, $\mathbb{C}$ would disagree with $\mathbb{R}$ on the statement $\exists x (x^2=-1)$. One start with $\mathbb{R}$ and imposes a solution $i$ of the polynomial $x^2+1=0$ to get $\mathbb{C}=\mathbb{R}[i]$.

The beginner is referred to [2] and [5] for an elementary introduction to forcing. More details can be found in the textbooks [3] and [6].. Forcing is also related to boolean valued models, see [10] for instance.

The Cohen forcing adds reals to the universe. In set theory, reals are identified with a countable binary sequence or countable sequence on $\omega$. Cohen forcing approaches the generic Cohen real by finite strings. There are other forcings which adjoin different types of reals. For instance, Mathias forcing [7]is defined in Section 2.

The rest of this article is organised as follows: Section 2 introduces the Mathias forcing and its properties; Section 3 deals with a variation of Mathias forcing and focuses on the difference with the usual Mathias forcing; Section 4 generalizes to uncountable space and proves the main theorem.

Definition 1  A pair $(s, S)$ is called a Mathias pair, if

1. $s$ is an strictly increasing function from some finite number $n$ to $\omega$.

2. $S$ is an infinite subset of $\omega$.

3. $\text{range}(s) \subseteq \min(S)$.

Definition 2   Mathias forcing consists of all Mathias pairs with the partial order defined as: $(s, S)\le (t, T)$ if $s$ end extends $t$, $\text{range}(s) \setminus \text{range}(t) \subseteq T$ and $S \subseteq T$.

For a generic filter $G$, the Mathias real is defined as $g=\bigcup\{t: \exists T (t, T) \in G\}$, which is an increasing function from $\omega$ to $\omega$ in the generic extension.

Definition 3   A real $g$ dominates a real $f$, if $\exists m\in \omega \; \forall n>m \; (f(n)<g(n))$.

Definition 4   $g \in \omega^\omega \cap \mathbb{V}[G]$ is dominating over $\mathbb{V}$ if $\forall f \in \omega^\omega \cap \mathbb{V}$, $g$ dominates $f$.

Definition 5   $g \in \omega^\omega \cap \mathbb{V}[G]$ is unbounded by $\mathbb{V}$ if $\forall f \in \omega^\omega \cap \mathbb{V}$, $g$ is not dominated by $f$.

It is clear that a dominating real is unbounded. It can be proved that a Mathias real is dominating over the ground model, while a Cohen real is unbounded but not dominating.

In this section, we consider a stronger version of the Mathias forcing.

Consider the strong Mathias forcing $\mathbb{P}_0$ which consists of all Mathias pairs. $(s, S)\le (t, T)$ if $s$ end extends $t$, $\text{range}(s) \setminus \text{range}(t) \subseteq T$, $S \subseteq T$ and $T\setminus S$ is finite.

Namely, we enhance the usual Mathias forcing with the last requirement. As with the Mathias forcing, we can define the typical generic real $g=\bigcup\{t: \exists T (t, T) \in G\}$.

In contrast to Mathias forcing, the typical generic real of $\mathbb{P}_0$ is not a dominating real.

Theorem 6  $g$ is not dominating over the ground model.

Proof  A key observation is that, below any condition, there are exactly $\aleph_0$ many conditions. So this forcing is in a sense a disjoint union of $\mathfrak c$ many Cohen forcings.

Suppose $g$ is dominating, let $r \Vdash \dot g$ is dominating. List the conditions below $r$ as $\{p_n:n\in \omega\}$. Let $q_n \le p_n$ decide $g(n)$. Namely, let $k_n \in \omega$ be such that $q_n \Vdash \dot g(n)=k_n$. Define in the ground model that $f:\omega \rightarrow \omega $, $f(n)=k_n$. Since $r \Vdash \exists m \forall n>m (\dot g(n )> f(n))$. There is some $r' \le r$ and $m$ such that $\forall n>m \; r' \Vdash \dot g(n )> f(n)$. By the observation above, there is some $n>m$ such that $p_n \le r'$. Then $q_n \le r'$. Therefore,

$q_n \Vdash \dot g(n)=k_n \wedge \dot g(n )> f(n).$

A contradiction. However, we still have the following.

Theorem 7  $g$ is unbounded over the ground model.

Proof  Suppose otherwise. Let $(t, T) \Vdash f$ dominate $\dot g$. Moreover, by passing to a stronger condition, we can assume without loss of generality that

$\exists m \forall n>m \; (t, T) \Vdash f(n)>\dot g(n).$

Now we let $n=\max\{m+1, |t|\}$ and extend $(t, T)$ to $(s, S)$ such that $|s|>n $ and $s(n) \ge f(n)$. Since $(s, S) \Vdash \dot g(n)=s(n) $, we get a contradiction.

In this section, we generalize the results to uncountable case. We will focus on the first uncountable cardinal $\omega_1$.

An $\omega_1$-real is a function from $\omega_1$ to $\omega_1$.

Definition 8   A pair $(s, S)$ is called a $\omega_1$-Mathias pair, if

1. $s$ is an strictly increasing function from some countable ordinal $\alpha$ to $\omega_1$.

2. $S$ is an uncountable subset of $\omega_1$.

3. $\text{range}(s) \subseteq \min(S)$.

Definition 9   Consider the strong $\omega_1$-Mathias forcing $\mathbb{P}_1$ which consists of all $\omega_1$-Mathias pairs. $(s, S)\le (t, T)$ if $s$ end extends $t$, $\text{range}(s) \setminus \text{range}(t) \subseteq T$, $S \subseteq T$ and $T\setminus S$ is countable.

Remark 10   We might study the $\omega_1$-Mathias forcing. The generic $\omega_1$-real added by $\omega_1$-Mathias forcing is dominating. However, the question that whether reals were added is more complicated and we will not explain in this article.

Now we begin to study the properties of $\mathbb{P}_1$.

The main theorem we wish to prove is the following.

Theorem 11   The forcing $\mathbb{P}_1$ adds no reals; the generic $\omega_1$-real is unbounded. If $\text{CH}$ holds, then the generic $\omega_1$-real is not dominating.

Chain condition is usually a key to study a forcing. Set theoretic topologist usually prefer the forcing with countable chain condition (c.c.c.), as Martin's Axiom can be applied there. However, we have the following lemma.

Lemma 12   $\mathbb{P}_1$ does not satisfy the countable chain condition.

Proof  Fix a partition $\omega_1=\bigcup\{I_\alpha: \alpha <\omega_1\}$, with each $I_\alpha$ uncountable. Then the $(\emptyset, I_\alpha)$'s are mutually incompatible forcing notions.

The chain conditions plays an important role in preserving cardinal. Since the failure of c.c.c., one would ask if $\omega_1$ is preserved.

Fortunately, a closure property can be expected for $\mathbb{P}_1$.

Theorem 13   $\mathbb{P}_1$ is $\omega$-closed.

Proof  Let $(s_0, S_0)\ge(s_1, S_1)\ge\cdots \ge (s_n, S_n)\ge \cdots$ be a decreasing sequence of conditions. Define $s=\bigcup\{s_n:n<\omega\}$. Then $s$ is an increasing function into $\omega_1$. Let $ \text{domain}(s_n)=\alpha_n$, $ \text{domain}(s)=\alpha$, then $\alpha=\sup\{\alpha_n:n\in \omega\}$ is a countable ordinal.

Let $S=\bigcap\{S_n:n<\omega\}$. Then $S=S_0 \setminus \bigcup \{S_n\setminus S_{n+1}: n \in \omega\}$. So $S$ is uncountable.

Since for each $n$, $\text{range}(s_n) \subseteq \min(S_n) \le \min(S)$, $\text{range}(s) \subseteq \min(S)$.

Therefore, $(s, S)$ is a condition. Since for each $n$, $S_n \setminus S =\bigcup\{S_k \setminus S_{k+1}: k\ge n \}$ is a countable set, $(s, S)$ is a lower bound for the given decreasing sequence.

It is well known that $\omega$-closed forcing does not add countable sequence of ordinals. In other words, it is $\omega$-distributive. See [3], for instance. So $\omega_1$ is preserved. Also, we have

Corollary 14   $\mathbb{P}_1$ does not add a real.

$\omega$-closed forcing is proper, see [8]. The class of forcing notions that are both $\omega$-distributive and proper was studied in [9], which focus on the forcing axiom and preservation of a couple of combinatorial properties. An immediate consequence is the following.

Corollary 15   Assume $\text{CH}$ in the ground model, then in a generic extension by $\mathbb{P}_1$, or a countable support iteration of $\mathbb{P}_1$, all the following cardinal invariants are equal to $\aleph_1$

$ \mathfrak a, \mathfrak b, \mathfrak d, \mathfrak e, \mathfrak g, \mathfrak h, \mathfrak i, \mathfrak m, \mathfrak p, \mathfrak r, \mathfrak s, \mathfrak t, \mathfrak u, \\ {\rm add}(\mathcal{B}), {\rm cov}(\mathcal{B}), {\rm non}(\mathcal{B}), {\rm cof}(\mathcal{B}), {\rm add}(\mathcal{L}), {\rm cov}(\mathcal{L}), {\rm non}(\mathcal{L}), {\rm cof}(\mathcal{L}). $

Remark 16   This result is in contrast to the case of usual Mathias forcing. With a countable support iteration of the usual Mathias forcing, $ \mathfrak e, \mathfrak m, \mathfrak p, \mathfrak t, {\rm add}(\mathcal{B}), {\rm cov}(\mathcal{B}), {\rm add}(\mathcal{L}), $ ${\rm cov}(\mathcal{L}) $ remain $\aleph_1$ while the rest become $\mathfrak c$.

As before, if $G$ is a generic filter of $\mathbb{P}$, then let $g=\bigcup\{t: \exists T (t, T) \in G\}$. $g$ is a generic $\omega_1$-real.

Unboundedness and dominating properties can be defined for a generic $\omega_1$-real, similar to Definitions 3 to 5.

Theorem 17   $g$ is an unbounded $\omega_1$-real over the ground model.

Proof  The proof is similar to the proof of Theorem 7.

Theorem 18   Assume $\text{CH}$ holds in the ground model, then $g$ is not dominating over the ground model.

In fact, we can prove a stronger theorem.

Theorem 19   Assume for any condition $r$ in a forcing $\mathbb{Q}$, $\{q\in \mathbb{Q}: q \le r\}$ has cardinality $\aleph_1$. Then for any $\omega_1$-real $h$ in the generic extension, there is in the ground model an $\omega_1$-real $f$, which is not dominated by $h$.

Proof  The proof is similar to that of Theorem 6. Suppose the conclusion fails. Then there is some $r \in \mathbb{Q}$ such that $r \Vdash \dot h$ is a dominating $\omega_1$-real.

List $\{r' \in\mathbb{Q}: r'\le r\} =\{p_\alpha: \alpha < \omega_1\}$. For each $\alpha$, let $q_\alpha \le p_\alpha$, $\beta_\alpha <\omega_1$ be such that $q_\alpha \Vdash \dot h(\alpha)=\beta_\alpha$. Now define in the ground model an $\omega_1$-real $f$ with $f(\alpha)=\beta_\alpha$.

Since $r \Vdash f$ is dominated by $h$, there is some $p \le r$ and $\gamma < \omega_1$ such that $p \Vdash \forall \alpha >\gamma \; (f(\alpha)< \dot h(\alpha))$. Since there are uncountably many successor of $p$, find some $\alpha > \gamma$ such that $p_\alpha \le p$. Then $q_\alpha \le p $ and $q_\alpha \Vdash \dot h(\alpha)=f(\alpha)$. A contradiction.

Proof of Theorem 18  Assume $\text{CH}$ holds in the ground model. Obviously, for any given condition $(t, T)$ of $\mathbb{P}_1$, the cardinality of the set $\{(t', T')\in \mathbb{P}_1: (t', T')\le (t, T)\}$ is

$|\omega_1^{<\omega_1}\times \omega_1^{<\omega_1}|=|\omega_1^{<\omega_1}|=\aleph_1 \cdot |\omega_1^{ \omega}|=\aleph_1 \cdot \aleph_1 \cdot \aleph_0^{\aleph_0}=\aleph_1.$

The second equality is due to the Hausdorff's formula; the last equality is $\text{CH}$.

Thus Theorem 19 applies and we conclude that the forcing $\mathbb{P}_1$ adds an unbounded $\omega_1$-real but adds no dominating $\omega_1$-real.

Remark  The results in this section can be generalize to any regular cardinal $\kappa$ with $\kappa^{<\kappa}=\kappa$.