#paper #forcing #forcing_axioms # Abstract $\diamondsuit$, the existence of gap-1 morasses, and the existence of more complicated morasses with built-in $\diamondsuit$ sequences all have equivalent formulations in terms of the existence of generic sets for partial orders, i.e. forcing axioms. > [!info] Application > - To apply one of these theorems we simply have to describe a construction using a partial order, and usually this is easier than actually carrying out the construction. > - This kind of application actually gives us two theorems from just one argument. If we define a partial order and family of dense sets and show that they have certain properties, we can conclude that a sufficiently generic set e:dsts in L, and also that such a generic set can be added to a model of ZFC by forcing. > - In some cases we can avoid an iterated forcing argument by assuming V = L and applying one of these theorems to several different partial orders. Since we never leave our ground model L, we can prove the existence of generic sets for several partial order,~: without having to worry about the complications involved in iterated forcing. # 4. $\diamondsuit$ ![[Diamond#^99868e]] Let $\mathcal{L}$ be a language with counably many symbols of all types. **Theorem 4.1.3** $\diamondsuit_{\kappa}$ is equivalent to: There is a sequence of $\mathcal{L}$ structures $\left< \mathfrak{A}_{\alpha}\mid \alpha<\kappa \right>$ such that $\forall \alpha<\kappa(A_{\alpha}=\alpha)$ and for any $\mathcal{L}$-structure $\mathfrak{A}$ with universe $A=\kappa$, $\exists \alpha<\kappa(\mathfrak{A}_{\alpha}\prec\mathfrak{A})$. >[!note]+ Notations >Fix a $\kappa$-closed poset $\mathbb{P} = (P, \leq )$ and a family of dense open sets $\mathcal{D}=\{ D_{\alpha}\mid \alpha<\kappa \}$. >- For $p\in P$, $\mathrm{rlm}(p)=\{ \alpha<\kappa \mid p \in D_{\alpha} \}$ >- $P_{\alpha}=\{ p \in P \mid \mathrm{rlm}(p) \subseteq \alpha \}= \{ p\in P \mid p\in D_{\beta} \implies \beta<\alpha\}$ >- Suppose $F \subseteq P$. The dense area of $F$, is $\mathrm{dns}(F)=\{p \in P \mid \exists q \in F (q \leq p)\}$. Upward closure of $F$. > Note that if $F$ is dense in $\mathbb{P}$, then $\mathrm{dns} ( F ) = P$. >- Suppose $Q,F \subseteq P$. $\mathrm{dns}^Q(F)=\{p \in Q \mid \exists q \in Q\cap F (q \leq p)\}$ >- If $\mathfrak{A}$ is an $\mathcal{L}$-structure with universe $\kappa$ , we will say $\mathfrak{B}\prec_{O}\mathfrak{A}$ ($\mathfrak{B}$ is an ordinal substructure of $\mathfrak{A}$) iff $\mathfrak{B}\prec \mathfrak{A}$ and $B$ is an ordinal less than $\kappa$. **Theorem 4.1.6.** $\diamondsuit_{\kappa}$ is equivalent to: Let $\mathbb{P} = (P, \leq )$ a poset and $\mathcal{D}=\{ D_{\alpha}\mid \alpha<\kappa \}$ a family of dense open sets. Suppose: 1. $\forall \alpha<\kappa$ $P_{\alpha}$ is lt;\kappa$-closed and $P=\bigcup_{\alpha<\kappa}P_{\alpha}$. 2. For each $\mathcal{L}$-structure $\mathfrak{B}$ with universe an ordinal $\alpha<\kappa$ we have a set $F_{\mathfrak{B}}\subseteq P$ such that $\{p \in P \mid \mathrm{rlm}(p) = \alpha\} \subseteq \mathrm{dns}(F_{\mathfrak{B}})$. Then there is a set $G$ which is $\mathbb{P}$-generic over $\mathcal{D}$, and for any $\mathcal{L}$-structure $\mathfrak{A}$ with universe $\kappa$, $\exists \mathfrak{B}\prec_O \mathfrak{A}$ s.t. $G\cap F_{\mathfrak{B}}\ne \varnothing$. Furthermore, $G$ can be chosen to be $\kappa$-complete. >[!info]+ Remark > The set $D_{\alpha}$ correspond to conditions giving the construction up to $\alpha$. > $\begin{align*} \{p \in P \mid \mathrm{rlm}(p) = \alpha\} &=\\ \{ p \in P \mid p\in D_{\beta} \iff \beta<\alpha \} &=\\ \bigcap_{\beta<\alpha} D_{\beta} \setminus \bigcup_{\alpha\leq\beta}D_{\beta} &\subseteq \mathrm{dns}(F_{\mathfrak{B}}) \end{align*}$ > The way we use it is that if $p$ corresponds to a construction up to $\alpha$, then it can be extended with a condition from $F_{\mathfrak{B}}$ for any $\mathfrak{B}$ with universe $\alpha$. >[!info]+ Remark >Note that for lt;\kappa$-closed posets, meeting $\kappa$ many dense sets with a $\kappa$-complete filter is ensured by AC (even $\mathrm{DC}_{\kappa}$). See [[Viale - Useful axioms#Forcing axioms]] **Lemma 4.2.2** If $2^{<\kappa}=\kappa$ then there is a suborder $\mathbb{Q}$ of $\mathbb{P}$ with $\lvert Q \rvert=\kappa$ having the same properties . **Theorem 4.2.3** If $2^{<\kappa}=\kappa$ and $\mathbb{P},\mathcal{D},F_{\mathfrak{B}}$ are as in the theorem, then there is a generic extension $V[H]$ ($H$ is generic for $\mathbb{Q}$) preserving cofinalities, cardinalities and exponentiation, where there is a $\mathbb{P}$-generic $G$ as in thm 4.1.6. **Theorem 4.2.4.** Assume $2^{<\kappa}=\kappa$ and let $\mathbb{P} = (P, \leq )$ is a lt;\kappa$-closed poset of size $\kappa$ and $\mathbb{Q}$ the poset of partial functions from $\kappa$ to $\kappa$ of size lt;\kappa$, which adds a $\diamondsuit_{\kappa}$-sequence. Then for $G\subseteq P$ generic, eithery $V[G]=V$ or $V[G]=V[H]$ for some $H\subseteq Q$ generic. ^d817b6 ### Constructing Souslin tree **Theorem 4.2.4** $\diamondsuit_{\omega_{1}}$ implies that there is an $\omega_{1}$-Souslin tree **Pf sketch.** $P=\{ \left< \tau,\leq \right> \mid \tau <\omega_{1} \text{ and } \left< \tau,\leq \right> \text{ is a normal tree} \}$ Ordered by end-extension. $D_{\alpha}=\{ p \in P \mid \alpha<\tau^{p} \}$ +1 Note $\mathrm{rlm}(p)=\tau^{p}$ and $P_{\alpha}=\{ p\in P \mid \tau^{p}\subseteq \alpha \}$ Fix a unary predicate symbol $\mathbf{X}$, and for every model $\mathfrak{B}$ with universe $\tau$, let $F_{\mathfrak{B}}=F^{0}_{\mathfrak{B}}\cup F^{1}_{\mathfrak{B}}$ where $F^{0}_{\mathfrak{B}}=\{ p\in p \mid \mathrm{rlm}(p)=\tau \text{ and } \mathbf{X}^{\mathfrak{B}} \text{ is not a maximal antichain in }\left< \tau^{p},\leq^{p} \right> \}$ $F^{1}_{\mathfrak{B}}=\{ p\in p \mid \exists \alpha \text{ s.t. every point at level } \alpha \text{ in the tree }\left< \tau^{p},\leq^{p} \right> \text{ is above some element of }\mathbf{X}^{\mathfrak{B}} \}$ - The sets $D_{\alpha}$ ensure the generic gives a tree of height $\omega_{1}$. - All levels are countable by the choice of the relation as end-extension. - $F^{1}_{\mathfrak{B}}$ ensures that antichains are countable - given a maximal antichain $X$, take a model $\mathfrak{A}$ where $\mathbf{X}^{\mathfrak{A}}=X$ and $\leq^{\mathfrak{A}}=\leq_{T}$ . The theorem tells us that there are $\mathfrak{B}\prec_{O}\mathfrak{A}$ and $p\in G\cap F_{\mathfrak{B}}$. We will get that in fact $p \in F^{1}_{\mathfrak{B}}$ and then get that $X=X \cap \tau^{p}$ . - $F^{0}_{\mathfrak{B}}$ is needed for the almost-denseness, i.e. for $\{p \in P \mid \mathrm{rlm}(p) = \alpha\} \subseteq \mathrm{dns}(F_{\mathfrak{B}})$.