## New location

May 24, 2013 at 13:53 | Posted in Uncategorized | 1 Comment

For those who may be interested, I’m migrating this blog to

http://eteisworth.blogspot.com/

because I can use mathjax more easily over there.

## The Story of Generators – 3 (Existence)

The Exact Upper Bound Argument

Let us assume that ${A}$ is as usual, with ${\lambda\in{\rm pcf}(A)}$. If we want to produce a generator for ${\lambda}$, we saw last time that what we need is a universal sequence for ${\lambda}$ that in addition possesses an exact upper bound modulo ${J_{<\lambda}[A]}$.

I don’t want to make an excursion into the theory of exact upper bounds, as there have been many high quality write-ups of this material already: in addition to Section 2.1 of the Abraham/Magidor article, there are also some materials due to Kojman that do an excellent job of exposing this material. The point is that one can use results of Shelah to modify the universal sequence so that it ends up with an exact upper bound, and then we get generators by the previous post.

Instead of re-hashing such arguments, I offer the following direct route from universal sequences to generators. The proof takes advantage of the ideal ${I[\lambda]}$, and we will see similar arguments shortly when we talk about transitivity of generators. The argument is based on the exposition of generators given in the Burke-Magidor pcf paper. It turns out that framing things in terms of ${I[\lambda]}$ simplifies things considerably.

From universal sequences to generators

Theorem 1 Suppose ${A}$ is a progressive set of regular cardinals and ${\lambda\in{\rm pcf}(A)}$. Then there is a generator for ${\lambda}$.

Proof: We can assume that ${\lambda>|A|^{++}}$. (If either ${|A|^+}$ or ${|A|^{++}}$ are in ${A}$, then they have generators consisting of singletons.)

Let ${\bar{f}=\langle f_\alpha:\alpha<\lambda\rangle}$ be a universal sequence for ${\lambda}$, that is, a sequence such that

• ${\alpha<\beta\Longrightarrow f_\alpha<_{J_{<\lambda}[A]} f_\beta}$, and
• ${\langle f_\alpha:\alpha<\lambda\rangle}$ is cofinal in ${\prod A/D}$ whenever ${D}$ is an ultrafilter on ${A}$ satisfying ${{\rm cf}(\prod A/D)=\lambda}$.

Such sequences exist by Theorem 4.2 of the Abraham/Magidor article, or see Lemma 2.1 on page 327 of Cardinal Arithmetic.

Let ${\kappa=|A|^+}$. Since we have assumed ${\kappa^+<\lambda}$, we know there is a stationary set ${S\subseteq S^\lambda_\kappa}$ lying in the ideal ${I[\lambda]}$. (See the first section of [Sh:420], or Theorem 3.18 in my own Handbook article.)

This means that there is a family ${\bar{C}=\langle C_\alpha:\alpha<\lambda\rangle}$ and a club ${E\subseteq\lambda}$ such that

• ${C_\alpha}$ is a closed (possibly bounded) subset of ${\alpha}$
• if ${\beta\in{\rm nacc}(C_\alpha)}$ then ${C_\beta=C_\alpha\cap\beta}$, and
• if ${\delta\in E\cap S}$ then ${C_\delta}$ is club in ${\delta}$ of order-type ${\kappa}$.

For each ${\alpha<\lambda}$ with ${C_\alpha\neq\emptyset}$, we define

$\displaystyle f^*_\alpha=\sup\{f_\beta:\beta\in C_\alpha\}. \ \ \ \ \ (1)$

If ${C_\alpha=\emptyset}$, we set ${f^*_\alpha=f_\alpha}$.

Since ${\prod A/J_{\leq\lambda}[A]}$ is ${\lambda^+}$-directed, we can find a single function ${h}$ bounding the collection ${\{f^*_\alpha:\alpha<\lambda\}}$ modulo ${J_{\leq\lambda}[A]}$.

Now we let ${N}$ be an elementary submodel of ${H(\chi)}$ for some sufficiently large regular cardinal ${\chi}$ such that

• ${A}$, ${\bar{f}}$, ${\bar{C}}$, ${E}$, ${S}$, ${h}$, and ${\lambda}$ are all in ${N}$
• ${|N|<\lambda}$
• ${N\cap\lambda}$ is an ordinal ${\delta\in S}$

This is possible because ${S}$ is stationary in ${\lambda}$.

We define

$\displaystyle B:=\{\theta\in A: h(\theta)

We prove that ${B}$ is a generator for ${\lambda}$. Part of this is simple, as by choice of ${h}$ we know ${B\in J_{\leq\lambda}[A]}$. To finish, we must establish the following:

Proposition 2 If ${D}$ is an ultrafilter on ${A}$ with ${{\rm cf}(\prod A/D)=\lambda}$, then ${B\in D}$.

The proof is not difficult, but the following lemma is critical.

Lemma 3 ${B\in N}$.

Proof:

This is where the ${I[\lambda]}$ assumption pays dividends. Note that ${\delta\in E}$ as ${E\in N}$, and therefore we know that ${C_\delta}$ is club in ${\delta}$ with order-type ${\kappa}$.

Given ${\alpha<\beta}$ in ${{\rm nacc}(C_\delta)}$, we know that ${C_\alpha= C_\delta\cap\alpha}$ is an initial segment of ${C_\beta= C_\delta\cap\beta}$, and both are initial segments of ${C_\delta}$. Thus

$\displaystyle f^*_\alpha\leq f^*_\beta\leq f^*_\delta. \ \ \ \ \ (3)$

Given ${\theta\in A}$ and ${\xi, we know there is a ${\gamma\in C_\delta}$ with ${\xi. Since ${C_\delta}$ is cofinal in ${\delta}$, we know ${\min(C_\delta\setminus(\gamma+1))}$ is an ordinal ${\alpha}$ in ${{\rm nacc}(C_\delta)}$ and ${\gamma\in C_\alpha= C_\delta\cap\alpha}$. Thus ${\xi, and we see

$\displaystyle f^*_\delta=\sup\{f^*_\alpha:\alpha\in{\rm nacc}(C_\delta)\}. \ \ \ \ \ (4)$

In particular, if ${h(\theta), then there is an ${\alpha(\theta)<\delta}$ such that

$\displaystyle \alpha\in {\rm nacc}(C_\delta)\setminus\alpha(\theta)\Longrightarrow h(\theta)

Since ${C_\delta}$ is club in ${\delta}$ with order-type ${\kappa>|A|}$, we can find a single ${\alpha\in{\rm nacc}(C_delta)}$ such that for all ${\theta\in A}$,

$\displaystyle h(\theta)

In particular,

$\displaystyle B=\{\theta\in A: h(\theta)

The set on the right is definable from parameters available in ${N}$ (both ${h}$ and ${f^*_\alpha}$ are there), and therefore ${B\in N}$. $\Box$

Let us return now to the proof of Proposition 2.

Proof: By way of contradiction, suppose ${D}$ is an ultrafilter on ${A}$ forming a counterexample. Since ${B}$ is in ${N}$, we can assume that ${D}$ is in ${N}$ as well (this is the key point).

This sequence ${\bar{f}}$ is universal for ${\lambda}$ and therefore the collection ${\{f_\alpha:\alpha<\lambda\}}$ is cofinal in ${\prod A/D}$. Since ${D\cap J_{<\lambda}[A]=\emptyset}$, we know furthermore that the sequence ${\langle f_\alpha:\alpha<\lambda\rangle}$ is ${<_D}$ increasing.

Thus, there is an ${\alpha^*<\lambda}$ such that ${h<_D f_\alpha}$ whenever ${\alpha^*\leq\alpha<\lambda}$. Again, we may assume that ${\alpha^*}$ is in ${N}$ (and hence less than ${\delta=N\cap\lambda}$).

Choose ${\beta\in C_\delta}$ greater than ${\alpha^*}$ and let ${\alpha=\min(C_\delta\setminus(\beta+1)}$. Then ${\alpha\in{\rm nacc}(C_\delta)}$ and ${C_\alpha= C_\delta\cap\alpha}$. In particular, ${\alpha\in N}$ and ${\beta\in C_\alpha}$

So we have

$\displaystyle h<_D f^*_\alpha\leq f^*_\delta. \ \ \ \ \ (8)$

One the other hand, since ${B\notin D}$ we know

$\displaystyle f^*_\delta\leq_D h. \ \ \ \ \ (9)$

But now we have obtained a contradiction. $\Box$

Proposition 2 taken together with the fact that ${B\in J_{\leq\lambda}[A]}$ easily establishes that ${B}$ is a generator for ${\lambda}$, so we are done. $\Box$

## The Story of Generators – 2

Finding Generators

Theorem 1 Suppose

• ${A}$ is a progressive sequence of regular cardinals
• ${\lambda\in{\rm pcf}(A)}$
• ${\bar{f}=\langle f_\alpha:\alpha<\lambda\rangle}$ is a universal sequence for ${\lambda}$
• ${g}$ is an exact upper bound for ${\bar{f}}$ modulo ${J_{<\lambda}[A]}$
• ${B=\{\theta\in A:g(\theta)=\theta\}}$

Then for any ultrafilter ${D}$ on ${A}$,

$\displaystyle {\rm cf}(\prod A/D)=\lambda \text{ if and only if } B\in D \text{ and }D\cap J_{<\lambda}[A]=\emptyset. \ \ \ \ \ (1)$

What does this have to do with generators? Just note the following:

Corollary 2 Under the assumptions of the theorem, we have ${J_{\leq\lambda}[A]=J_{<\lambda}[A]+B}$, hence ${B}$ is a generator for ${\lambda}$.

Proof: If ${D}$ is an ultrafilter on ${A}$ containing ${B}$, then either ${D}$ meets ${J_{<\lambda}[A]}$ or it does not. In the first case, the cofinality of ${\prod A/D}$ is less than ${\lambda}$ by definition of ${J_{<\lambda}[A]}$, and in the second case the cofinality is exactly ${\lambda}$ by the conclusion of the theorem. In either case, the cofinality is at most ${\lambda}$ and so ${B}$ is in ${J_{\leq\lambda}[A]}$ and we have

$\displaystyle J_{<\lambda}[A]+B\subseteq J_{\leq\lambda}[A]. \ \ \ \ \ (2)$

For the other inclusion, suppose ${B^*}$ is in ${J_{\leq\lambda}[A]}$ but not in ${J_{<\lambda}[A]+B}$. Let ${D}$ be an ultrafilter on ${A}$ containing ${B^*}$ but disjoint to ${J_{<\lambda}[A]+B}$. Since the cofinality of ${\prod A/D}$ is ${\lambda}$, we have contradicted the conclusion of the theorem. $\Box$

Digression on exact upper bounds

Before proving Theorem~1, we need to say a few words about exact upper bounds because different authors treat them in slightly different ways. Let us assume that ${g}$ is an exact upper bound for ${\bar{f}}$ mod ${J_{<\lambda}[A]}$ just as in the statement of the theorem. It is easy to see

$\displaystyle \{\theta\in A: g(\theta)=0\text{ or }\theta

and so our ${g}$ is equal mod ${J_{<\lambda}[A]}$ to a function ${g^*}$ satisfying

$\displaystyle 0

If we define ${B^*:=\{\theta\in A:g^*(\theta)=\theta\}}$, then ${B^*}$ and ${B}$ are equal modulo the ideal ${J_{<\lambda}[A]}$, and for any ultrafilter ${D}$ on ${A}$ disjoint to ${J_{<\lambda}[A]}$, we have ${B\in D}$ if and only if ${B^*\in D}$.

The point of the above is that we can replace ${g}$ by ${g^*}$ and not change anything, so we may as well assume that our function ${g}$ satisfies

$\displaystyle 0

Proof of Theorem

Proof: Suppose first that ${D}$ is an ultrafilter on ${A}$ containing ${B}$ but disjoint to ${J_{<\lambda}[A]}$. The sequence ${\bar{f}}$ is ${<_D}$-increasing, so if we can show it is cofinal in ${\prod A/D}$ we will know that the cofinality of ${\prod A/D}$ is exactly ${\lambda}$.

Suppose ${h\in \prod A}$. Then by setting ${h}$ equal to zero outside of ${B}$ we produce a function ${h^*}$ that is equal to ${h}$ mod ${D}$ and less than ${g}$ everywhere. Our assumptions on ${g}$ then give us an ${\alpha}$ such

$\displaystyle h^*<_{J_{<\lambda}[A]} f_\alpha, \ \ \ \ \ (6)$

and since ${D\cap J_{<\lambda}[A]=\emptyset}$, we achieve

$\displaystyle h=_D h^*<_D f_\alpha, \ \ \ \ \ (7)$

as required.

For the other direction, suppose by way of contradiction that ${D}$ is an ultrafilter on ${A}$ satisfying ${{\rm cf}(\prod A/D)=\lambda}$ with ${B\notin D}$.

Outside of the set ${B}$, we have ${g(\theta)<\theta}$ and so ${g}$ is ${D}$-equivalent to a function ${g^*\in\prod A}$. Since ${\bar{f}}$ is universal for ${\lambda}$, there is an ${\alpha<\lambda}$ such that ${g^*<_D f_\alpha}$ and hence

$\displaystyle g<_D f_\alpha \ \ \ \ \ (8)$

as well.

But ${g}$ is an exact upper bound for ${\bar{f}}$ mod ${J_{<\lambda}[A]}$, and so

$\displaystyle f_\alpha\leq_{J_{<\lambda}[A]} g. \ \ \ \ \ (9)$

Since the cofinality of ${\prod A/ D}$ is ${\lambda}$ we know ${D\cap J_{<\lambda}[A]=\emptyset}$, and therefore

$\displaystyle f_\alpha\leq_D g. \ \ \ \ \ (10)$

Putting all of this together yields

$\displaystyle f_\alpha\leq_D g <_D f_\alpha, \ \ \ \ \ (11)$

and this is a contradiction. $\Box$

## The Story of Generators 1

Introduction

I’m not sure exactly where things left off, so I’ll just begin with a series of posts on generators for pcf. I will also try to keep the posts short so that I can ease back into the routine of writing them.

Let us assume that ${A}$ is a progressive set of regular cardinals, and ${\lambda\in{\rm pcf}(A)}$. I will assume we know already some of the basics about the ideal ${J_{<\lambda}[A]}$ and I will use these facts without much comment.

What we want to look at is the existence of a generator for ${\lambda}$: we will sketch the proof that the ideal ${J_{\leq\lambda}[A]}$ is generated over ${J_{<\lambda}[A]}$ by a single set ${B_\lambda}$. And I do mean “sketch”, as the details are worked out nicely in Section 4 of the Abraham/Magidor article in the Handbook of Set Theory.

Universal Sequences

Definition 1 Suppose ${\lambda\in{\rm pcf}(A)}$. A sequence ${\bar{f}=\langle f_\alpha:\alpha<\lambda\rangle}$ of functions in ${\prod A}$ is a universal sequence for ${\lambda}$ if

• ${\bar{f}}$ is ${<_{J_{<\lambda}[A]}}$-increasing, and

• ${\bar{f}}$ is cofinal in ${\prod A/D}$ whenever ${D}$ is an ultrafilter on ${A}$ with ${\lambda={\rm cf}(\prod A/D)}$.

Universal sequences are tightly related to the existence of generators for pcf, as we shall see. I want to point out the following result:

Theorem 2 Suppose ${A}$ is a progressive set of regular cardinals and ${\lambda\in{\rm pcf}(A)}$ Then the following statements are equivalent:

1. There is a universal sequence for ${\lambda}$.
2. There is a family ${F\subseteq \prod A}$ such that for any ultrafilter ${D}$, if ${\prod A/D}$ has cofinality ${\lambda}$ then ${F}$ remains unbounded in ${\prod A/D}$.
3. There is a family ${\langle B_\alpha:\alpha<\lambda\rangle}$ of subsets of ${A}$ such that
• ${\alpha<\beta\Longrightarrow B_\alpha\subseteq B_\beta}$ modulo ${J_{<\lambda}[A]}$, and
• ${J_{\leq\lambda}[A]}$ is the ideal generated by ${J_{<\lambda}[A]}$ together with the sets ${\{B_\alpha:\alpha<\lambda\}}$.

The above is basically Fact 2.2 on page 13 of Cardinal Arithmetic, and it follows quite easily from the work done in the first section of the book. (The 3rd statement says, in the notation of the book, that ${\lambda}$ is semi-normal.) Abraham and Magidor develop basic pcf theory in a slightly different order, and deriving the above result from the material they present prior to defining universal sequences is a bit more difficult.

Of course, the main point is the following result:

Theorem 3 If ${A}$ is a progressive set of regular cardinals, then every ${\lambda\in{\rm pcf}(A)}$ has a universal sequence.

Again, we will not prove the above as this is Theorem 4.2 of the Abraham-Magidor article, and their proof is quite clear.

What have we learned from the above? We have outlined the first steps towards proving that generators exist. Putting the two theorems presented here together, we see that if ${\lambda\in{\rm pcf}(A)}$, then ${J_{\leq\lambda}[A]}$ is pretty simply generated over ${J_{<\lambda}[A]}$.

In our next post, we’ll see how “tuning up” a universal sequence leads to the existence of generators.

## Miss me?

May 21, 2012 at 11:05 | Posted in Uncategorized | 1 Comment

Will be back posting this week. The long break was due to being busy with other aspects of my career, but things have settled back down and my sabbatical starts in three weeks.

## Dictionary

I took a look at the Abraham/Magidor article in the Handbook of Set Theory, and what they discuss in Section 5 of their article is within epsilon of what Shelah uses in Section 1 of Chapter VIII of The Book. There are minor technical differences,but their writing is so much clearer than Shelah’s that I’m tempted to prove the main results of Chapter VIII section 1 using their version of things (minimally obedient universal sequences and ${\kappa}$-presentable models) instead of Shelah’s (“suppose (a)-(e) of 1.2 hold”).

What this means in practical terms is that I’m going to be doing some translation of Section 1 of Chapter VIII into the language of the Handbook and see how well the proofs go through.

Tedious, but probably worthwhile in the interest of making the material accessible!

## Some Assumptions

I have been contemplating exactly where to begin the discussion of these matters, and I think that I want to go back to Chapter VIII of The Book, and look at the first section of it because the arguments there keep appearing in later works.

In this post, I’m going to write down some assumptions [(a)-(e) of Claim 1.2] that will be used in the next few posts.

The Assumptions

We assume the following

1. ${\mathfrak{a}}$ is a set of regular cardinals satisfying ${|\mathfrak{a}|^+<\min(\mathfrak{a})}$

2. For every ${\mathfrak{b}\subseteq\mathfrak{a}}$, we let ${\overline{f}^{\mathfrak{b}}=\langle f^{\mathfrak{b}}_\alpha:\alpha<\max{\rm pcf}(\mathfrak{b})\rangle}$ satisfy
• ${f^{\mathfrak{b}}_\alpha\in\prod\mathfrak{a}}$,
• ${\bar{f}^{\mathfrak{b}}}$ is strictly increasing modulo ${J_{<\max{\rm pcf}(\mathfrak{b})}[\mathfrak{a}]}$,
• if ${\alpha<\max{\rm pcf}(\mathfrak{b})}$ and ${|\mathfrak{a}|<{\rm cf}(\alpha)<\min(\mathfrak{a})}$, then for each ${\theta\in \mathfrak{b}}$ we have

$\displaystyle f^{\mathfrak{b}}_\alpha(\theta)=\min\{\bigcup_{\beta\in C}f^{\mathfrak{b}}_\beta(\theta): C\text{ club in }\alpha\}. \ \ \ \ \ (1)$

• for every ${f\in\prod\mathfrak{b}}$ and ${\alpha<\max{\rm pcf}(\mathfrak{b})}$, there is a ${\beta>\alpha}$ such that ${f (everwhere)

3. ${\chi}$ is a sufficiently large regular cardinal, ${<_\chi}$ is a well-ordering of ${H(\chi)}$

4. ${\bar{N}=\langle N_i:i\leq\delta\rangle}$ is an increasing continuous sequence of elementary submodels of ${\langle H(\chi), \in, <_\chi\rangle}$ such that
• ${|N_i|<\min(\mathfrak{a})}$
• ${\langle N_j:j\leq i \rangle \in N_{i+1}}$
• ${|N_i|+1\subseteq N_i}$
• ${\bar{f}=\{\bar{f}^{\mathfrak{b}}:\mathfrak{b}\subseteq\mathfrak{a}\}\in N_0}$

5. ${\mathfrak{a}\subseteq N_0}$ and ${|\mathfrak{a}|<{\rm cf}(\delta)\leq\delta<\min(\mathfrak{a})}$.

I’m going to do some “dictionary work” to get the official names for such objects. I just want to check which terms have become standard, and I’ll use the Abraham/Magidor Handbook article as the final word.

## Quick update

Well, I think hitting the medium-term goals is going to be much harder more interesting than I thought! I’ll start posting pieces of what I know starting next week after our Spring Break is over…

## Medium Term Goals

I feel the urge to return to the main work of this blog, namely working through the details of the more exotic portions of “The Book” and its continuations. I also happen to have an ideal body of results to attack:

As noted earlier, Shelah’s second proof of the “cov vs. pp Theorem” ([Sh:400] 3.5) contains an error (acknowledged in [Sh:513]), and although it is possible to effect a repair of most of it, the full version is still unproven. Now if we look ahead to [Sh:410] (one of the first papers continuing The Book) we find that a great many proofs of theorems end with a sentence saying roughly “now repeat the proof of [Sh:400, 3.5]” (I count at least four instances of this in Section 2 of the paper).

So on the face of it, it’s not clear how many of these results are actually valid. It may be that the weaker version of [Sh:400, 3.5] is strong enough to push through the arguments, but since there are very few details it’s hard to tell.

This looks like a nice task for me to tackle in the blog, so that’s probably what I will do.

So what does it mean when we say ${k_\beta}$ “almost works”? We’ll start with a lemma:

Lemma 1 For a fixed ${\nu\in^{<\omega}\mu}$, the set ${\{\alpha<\beta:\nu=\eta_\alpha\upharpoonright k_\beta(\alpha)+1\}}$ contains at most one element from each set ${A_{n+1}^\beta\setminus A^\beta_n}$, and hence is at most countable.

Proof: Suppose by way of contradiction what ${\alpha}$ and ${\gamma}$ are distinct members of ${A^\beta_{n+1}\setminus A^\beta_n}$ for which

$\displaystyle \nu=\eta_\alpha\upharpoonright (k_\beta(\alpha)+1)=\eta_\gamma\upharpoonright (k_\beta(\gamma)+1). \ \ \ \ \ (1)$

Then

$\displaystyle f^\beta_{n+1}(\alpha)=\eta_\alpha(k_\beta(\alpha))=\nu(k_\beta(\alpha))=\nu(k_\beta(\gamma))=\eta_\gamma(k_\beta(\gamma))=f^\beta_{n+1}(\gamma), \ \ \ \ \ (2)$

and this contradicts the fact that ${f^\beta_{n+1}}$ is a transversal for ${\{x_\epsilon:\epsilon\in A^\beta_{n+1}\}}$. $\Box$

Now given ${\alpha<\beta}$, we are going to define ${E(\alpha)}$ to be those ${\gamma<\beta}$ for which ${k_\beta}$ has failed to work, that is,

$\displaystyle E(\alpha)=\{\gamma<\beta:\max\{k_\beta(\alpha),k_\beta(\gamma)\}<\Delta(\alpha,\gamma)\}. \ \ \ \ \ (3)$

Lemma 2 The set ${E(\alpha)}$ is at most countable.

Proof: If not, find ${k^*<\omega}$ for which the set ${B:=\{\gamma\in E(\alpha):k_\beta(\gamma)=k^*\}}$ is uncountable, and set

$\displaystyle \nu=\eta_\alpha\upharpoonright k^*+1. \ \ \ \ \ (4)$

Then for each ${\gamma\in B}$, we have

$\displaystyle \eta_\gamma\upharpoonright k_\beta(\gamma)+1=\eta_\alpha\upharpoonright k^*+1=\nu, \ \ \ \ \ (5)$

which contradicts the preceding lemma. $\Box$

So for a given ${\alpha<\beta}$, the function ${k_\beta}$ will “disjointify” ${A_\alpha}$ from all but countably many ${A_\gamma}$ with ${\gamma<\beta}$: the function ${k_\beta}$ “almost works”.

Notice that ${\gamma\in E(\alpha)}$ if and only if ${\alpha\in E(\gamma)}$, so that we can define a graph ${\Gamma}$ on ${\beta}$ by connecting ${\alpha}$ and ${\gamma}$ if and only if ${\gamma\in E(\alpha)}$. By the preceding lemma, every vertex of the graph has at most countably many edges coming out of it. An easy argument tells us that the connected components of our graph ${\Gamma}$ are at most countable as well.

Now if ${\alpha}$ and ${\gamma}$ are members of different connected components of ${\Gamma}$ then ${k_\beta}$ will disjointify ${A_\alpha}$ and ${A_\gamma}$. But each connected component of ${\Gamma}$ can be disjointified easily (by induction) as it is at most countable.

Thus, it is straightforward now to “correct” ${k_\beta}$ to a function ${h_\beta}$ which will work everywhere.

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