## Finishing up Claim 3.3

So why does ${H}$ work? Let us define

$\displaystyle N^+:={\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup H); \ \ \ \ \ (1)$

we should show

$\displaystyle N\cap\mu\subseteq N^+. \ \ \ \ \ (2)$

It should come as no surprise that we’re going to accomplish this using our prior postings on elementary submodels. In particular, we need to jump back to our posting of May 24, where we proved a version of Shelah’s Claim 3.3A. The notation and hypotheses are slightly different in our current setting, and since last May is long time ago, I’ll just formulate another version of things and then prove it.

Lemma 1 Suppose ${\mu}$ is a singular cardinal and ${\mathfrak{A}}$ is as usual. Suppose we have objects ${\kappa}$, ${N}$, and ${H}$ such that

• ${N}$ is an elementary submodel of ${\mathfrak{A}}$,
• ${\kappa}$ is a cardinal less than ${\mu}$ and in ${N}$,
• ${H\subseteq N\cap \prod({\sf Reg}\cap(\kappa,\mu))}$, and
• for every ${\eta\in N\cap {\sf Reg}\cap(\kappa,\mu)}$ and ${\alpha\in N\cap\eta}$, there is an ${f\in H}$ such that ${f(\eta)>\alpha}$.

Then

$\displaystyle N\cap\mu\subseteq N^+:={\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup H). \ \ \ \ \ (3)$

Proof: Suppose this fails, and define

$\displaystyle \gamma(*)=\min(N\cap\mu\setminus N^+), \ \ \ \ \ (4)$

and

$\displaystyle \beta(*)=\min(N^+\setminus\gamma(*)). \ \ \ \ \ (5)$

It should be clear that ${\gamma(*)}$ is well-defined and ${\kappa<\gamma(*)<\mu}$. It is also obvious that ${\gamma(*)<\beta(*)}$. We will now show that ${\beta(*)<\mu}$.

This follows immediately if ${{\rm cf}(\mu)\leq\kappa}$, as then ${N^+\cap\mu}$ will be unbounded in ${\mu}$. If ${{\rm cf}(\mu)>\kappa}$, then

$\displaystyle {\rm cf}(\mu)\in N\cap{\sf Reg}\cap (\kappa,\mu), \ \ \ \ \ (6)$

and so ordinals of the form ${f({\rm cf}(\mu))}$ for ${f\in H}$ are cofinal in ${N\cap\mu}$. But each of these ordinals is also in ${N^+}$ (as ${{\rm cf}(\mu)}$ is in ${N^+}$ together with all members of ${H}$), and this tells us that ${N^+\cap\mu}$ must contain an ordinal bigger than ${\gamma(*)}$.

Thus, no matter what the cofinality of ${\mu}$, we know

$\displaystyle \kappa<\gamma(*)<\beta(*)<\mu. \ \ \ \ \ (7)$

Pausing for a moment, we note that the salient property of ${\beta(*)}$ guarantees

$\displaystyle N^+\cap [\gamma(*),\beta(*))=\emptyset. \ \ \ \ \ (8)$

This is quite powerful, given that we know ${\beta(*)\in N^+}$. For example, this immediately implies that ${\beta(*)}$ cannot be a successor ordinal, for example, as its predecessor would violate the above condition. Similarly, ${\beta(*)}$ must have cofinality greater than ${\kappa}$ — if not, then we can use the fact that ${\kappa\subseteq N^+}$ to derive a contradiction.

We’ll finish by showing that ${\kappa<{\rm cf}(\beta(*))}$ also leads to a contradiction. Given the definition of ${N^+}$, we know that any element of ${N^+\cap\mu}$ belongs to a subset of ${\mu}$ of cardinality at most ${\kappa}$ that is definable in ${\mathfrak{A}}$ from finitely many members of ${H}$. In particular, there is a set ${A}$ such that

• ${\beta(*)\in A}$,
• ${|A|\leq\kappa}$, and
• ${A}$ is definable in ${\mathfrak{A}}$ from finitely many members of ${H}$.

(We discussed this in more detail back in May.)

The definability of ${A}$ guarantees that it is an element of ${N}$ as well, and therefore ${\beta(*)\in N}$ as it can be defined as the minimal element of ${A\setminus\gamma(*)}$.

Thus ${\eta:={\rm cf}(\beta(*))}$ is an element of ${N\cap N^+}$. Let us fix a strictly increasing function ${h\in N\cap N^+}$ mapping ${\eta}$ onto a cofinal subset of ${\beta(*)}$. Let us define

$\displaystyle \delta(*):=h^{-1}(\{\gamma(*)\}). \ \ \ \ \ (9)$

Note that ${\delta(*)\in N\cap \eta}$. We have assumed ${\eta>\kappa}$, so this means

$\displaystyle \eta\in N\cap{\sf Reg}\cap (\kappa,\mu). \ \ \ \ \ (10)$

Thus our assumptions give us an ${f\in H}$ with ${\delta(*).

But ${f(\eta)}$ is an element of ${N^+}$ as well, hence so is the ordinal ${h(f(\eta))}$. We’ve now got a problem: since ${h}$ is an increasing function from ${\eta}$ into ${\beta(*)}$, it follows that

$\displaystyle \gamma^*=h(\delta(*))

contradicting the fact that ${N^+\cap [\gamma(*),\beta(*))=\emptyset}$. $\Box$

Now that we’ve established the lemma, let’s look back at what we need to verify in order to apply it. Looking at the bullet points in the statement of our lemma, all we need to worry about is the fourth one. Suppose ${\eta\in N\cap{\sf Reg}\cap (\kappa,\mu)}$ and ${\alpha\in N\cap\eta}$. Choose ${k}$ large enough so that

$\displaystyle \{\eta,\alpha\}\subseteq N_k. \ \ \ \ \ (12)$

Looking back at the construction, we then have

$\displaystyle \alpha\leq g_k(\eta)=\sup(N_k\cap\eta) \ \ \ \ \ (13)$

and our choice of ${H_{k}}$ guarantees the existence of a function ${f\in H_{k}\subseteq H}$ with

$\displaystyle \alpha\leq g_k(\eta)

Thus, the hypotheses of the lemma have been satisfied by the construction from the previous post, and we are finished.