## 580 -Cardinal arithmetic (12)

5. PCF theory

To close the topic of cardinal arithmetic, this lecture is a summary introduction to Saharon Shelah’s pcf theory. Rather, it is just motivation to go and study other sources; there are many excellent references available, and I list some below. Here I just want to give you the barest of ideas of what the theory is about and what kinds of results one can achieve with it. All the results mentioned are due to Shelah unless otherwise noted. All the notions mentioned are due to Shelah as far as I know.

Some references:

• Maxim Burke, Menachem Magidor, Shelah’s pcf theory and its applications, Annals of pure and applied logic, 50, (1990), 207–254.
• Thomas Jech, Singular cardinal problem: Shelah’s theorem on ${2^{\aleph_\omega}}$, Bulletin of the London Mathematical Society, 24, (1992), 127–139.
• Saharon Shelah, Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, 26 (2), (1992), 197–210.
• Saharon Shelah, Cardinal arithmetic, Oxford University Press, (1994).
• Menachem Kojman, The ABC of pcf, unpublished notes, available (as of this writting) at his webpage.
• Uri Abraham, Menachem Magidor, Cardinal arithmetic, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.
• Todd Eisworth, Successors of singular cardinals, in Handbook of set theory, Matthew Foreman, Akihiro Kanamori, eds., forthcoming.

Pcf came out of Shelah’s efforts to extend and improve the Galvin-Hajnal results on powers of singulars of uncountable cofinality (see lectures II.6 and II.7). His first success was an extension to cardinals of countable cofinality. For example:

Theorem 1

1. If ${\aleph_\omega}$ is strong limit, then ${2^{\aleph_\omega}<\aleph_{(2^{\aleph_0})^+}.}$
2. In fact, for any limit ordinal ${\xi,}$ ${\aleph_\xi^{|\xi|}<\aleph_{(2^{|\xi|})^+}.}$
3. If ${\delta}$ is limit and ${\delta=\alpha+\beta,}$ ${\beta\ne0,}$ then ${\aleph_\delta^{{\rm cf}(\delta)}<\aleph_{\alpha+(|\beta|^{{\rm cf}(\beta)})^+}.}$ ${\Box}$

Notice that this is an improvement of the Galvin-Hajnal bounds even if ${\xi}$ has uncountable cofinality. As impressive as this result may be, Shelah felt that it was somewhat lacking in that the bound was not “absolute” since ${2^{|\xi|}}$ can be arbitrarily large. The best known result of pcf theory remedies this issue:

Theorem 2 If ${\delta}$ is a limit ordinal and ${2^{|\delta|}<\aleph_{\alpha+\delta}}$ then ${\aleph_{\alpha+\delta}^{|\delta|}<\aleph_{\alpha+|\delta|^{+4}}.}$ ${\Box}$

These results are obtained by a careful study of the set products ${\prod{\mathfrak a}=\prod_{\kappa\in{\mathfrak a}}\kappa}$ for sets of cardinals ${{\mathfrak a}.}$ The basic idea is the following: Let ${\kappa\le\lambda}$ be cardinals, and let ${{\rm cov}(\lambda,\kappa)}$ be the cofinality of ${([\lambda]^\kappa,\subseteq),}$ i.e., the minimal size of a collection of subsets of ${\lambda}$ of size ${\kappa}$ such that any such subset is covered by one of the family:

$\displaystyle {\rm cov}(\lambda,\kappa)=\min\{|{\mathcal F}|:{\mathcal F}\subseteq [\lambda]^\kappa,\forall A\in[\lambda]^\kappa\exists B\in{\mathcal F}\,(A\subseteq B)\}.$

Then clearly ${\lambda^\kappa=2^\kappa+{\rm cov}(\lambda,\kappa):}$ On the one hand, ${\lambda^\kappa=|[\lambda]^\kappa|;}$ if ${{\mathcal F}}$ is cofinal in ${[\lambda]^\kappa,}$ then any subset of ${\lambda}$ of size ${\kappa}$ is in ${\bigcup\{[B]^\kappa:B\in{\mathcal F}\},}$ and this set has size at most ${\kappa^\kappa|{\mathcal F}|.}$ On the other hand, obviously ${2^\kappa+{\rm cov}(\lambda,\kappa)\le\lambda^\kappa.}$

Hence, to bound ${\lambda^\kappa,}$ at least if ${2^\kappa<\lambda,}$ reduces to understanding ${{\rm cov}(\lambda,\kappa).}$ Note that if ${2^\kappa\ge\lambda,}$ then trivially ${\lambda^\kappa<\aleph_{(2^\kappa)^+}.}$

More generally:

Definition 3 Let ${\lambda,\kappa,\theta,\sigma}$ be cardinals such that ${\lambda\ge\theta\ge\sigma,}$ ${\kappa\ge\aleph_0,}$ ${\theta>1,}$ ${\sigma>1,}$ and either ${\kappa\ge\theta,}$ or else (${\kappa^+=\theta}$ and ${{\rm cf}(\theta)<\sigma}$).

Let ${{\rm cov}(\lambda,\kappa,\theta,\sigma)}$ denote the least ${\mu}$ such that there is a family ${{\mathcal P}}$ of size ${\mu}$ of subsets of ${\lambda,}$ each of size less than ${\kappa,}$ such that whenever ${t\subseteq\lambda}$ and ${|t|<\theta,}$ there is some ${{\mathcal P}'\subseteq{\mathcal P}}$ with ${|{\mathcal P}|<\sigma}$ and ${t\subseteq\bigcup_{A\in{\mathcal P}'}A.}$

To understand these covering numbers, Shelah switches to the study of cofinalities of ${(\prod{\mathfrak a},\le_I)}$ for different ideals ${I;}$ so his results directly relate to the Galvin-Hajnal approach. The switch is justified in view of the following result (the ${{\rm cov}}$ vs ${{\rm pp}}$ theorem):

Definition 4 A partially ordered set ${(P,<)}$ has true cofinality ${\kappa}$ iff there is a ${<}$-increasing and cofinal sequence of length ${\kappa}$ of elements of ${P.}$ If this is the case, we write ${{\rm tcf}(P,<)=\kappa.}$

The definition above is nontrivial. For example, ${{\rm cf}(\omega\times\omega_1)=\omega_1,}$ where ${P=\omega\times\omega_1}$ is ordered by ${(n,\alpha)<(m,\beta)}$ iff ${n and ${\alpha<\beta.}$ However, there is no increasing ${\omega_1}$-sequence cofinal in ${P.}$ Similarly, we can arrange that ${{\rm cf}(P,<)}$ is singular, although this is impossible if ${P}$ has a true cofinality.

Definition 5 If ${{\rm cf}(\lambda)\le\kappa<\lambda}$ then the pseudopower ${{\rm pp}_\kappa(\lambda)}$ is the supremum of the true cofinalities of ${(\prod{\mathfrak a},<_I)}$ where ${{\mathfrak a}}$ is a set of at most ${\kappa}$ regular cardinals below ${\lambda}$ and unbounded in ${\lambda,}$ and ${I}$ varies among the ideals over ${{\mathfrak a}}$ extending the ideal of bounded sets such that ${{\rm tcf}(\prod{\mathfrak a},<_I)}$ exists.

${{\rm pp}(\lambda)}$ denotes ${{\rm pp}_{{\rm cf}(\lambda)}(\lambda)}$ and if ${\Gamma}$ is a class of ideals, then ${{\rm pp}_\Gamma(\lambda)}$ is defined just as ${{\rm pp}(\lambda),}$ but with only ideals in ${\Gamma}$ being considered. If ${J}$ is an ideal (extending the ideal of bounded sets), then ${{\rm pp}_J(\lambda)}$ denotes ${{\rm pp}_\Gamma(\lambda),}$ where ${\Gamma}$ is the class of ideals extending ${J.}$

Theorem 6 Let ${\Gamma=\Gamma(\theta,\lambda)}$ denote the set of (proper) ${\lambda}$-complete ideals over some cardinal ${\theta_I<\theta.}$

Suppose that ${\sigma}$ is regular, ${\sigma>\aleph_0,}$ and ${\lambda\ge\kappa\ge\theta>\sigma.}$ Then:

1. ${\sup\{{\rm pp}_\Gamma(\lambda^*):\lambda^*\in[\kappa,\lambda], \sigma\le{\rm cf}(\lambda^*)<\theta\}+\lambda}$ ${={\rm cov}(\lambda,\kappa,\theta,\sigma)+\lambda.}$
2. Moreover, if ${\mu={\rm cov}(\lambda,\kappa,\theta,\sigma)}$ is regular and larger than ${\lambda,}$ then for some ${I\in\Gamma(\theta,\sigma)}$ and ${(\lambda_\alpha:\alpha\in{\rm dom}(I)),}$ we have ${\mu\le{\rm tcf}(\prod_\alpha\lambda_\alpha,<_I)}$ and ${\theta<\lambda_\alpha\le\lambda.}$
3. In item 1, the inequality ${\le}$ holds without needing to require that ${\sigma>\aleph_0.}$
4. In item 1, if both sides of the equation are larger than ${\lambda,}$ then the sup is in fact a max. ${\Box}$

For a proof, see Theorem 5.4 of Shelah’s book. Shelah notes the following conclusions:

Corollary 7 Suppose that ${\lambda}$ is a singular cardinal of uncountable cofinality. Then

$\displaystyle {\rm cov}(\lambda,\lambda,({\rm cf}(\lambda))^+,{\rm cf}(\lambda))={\rm pp}_{\Gamma(({\rm cf}(\lambda))^+,{\rm cf}(\lambda))}(\lambda).\ \Box$

Lemma 8 Suppose that ${\lambda\ge\kappa\ge\theta>\sigma,}$ ${{\rm cf}(\kappa)\ge\sigma,}$ and either ${{\rm cf}(\theta)\ge\sigma,}$ or ${2^{<\theta}<\lambda.}$ Then

$\displaystyle \lambda^{<\theta}={\rm cov}(\lambda,\kappa,\theta,\sigma)^{<\sigma}+\sum_{\alpha<\kappa}|\alpha|^{<\theta}.\ \Box$

Corollary 9 Assume that ${{\rm cf}(\lambda)>\aleph_0.}$ Then

$\displaystyle \lambda^\theta={\rm cov}(\lambda,\lambda,({\rm cf}(\lambda))^+,{\rm cf}(\lambda))={\rm pp}_{\Gamma(({\rm cf}(\lambda))^+,{\rm cf}(\lambda))}(\lambda),$

unless ${\lambda\le2^\theta,}$ in which case ${\lambda^\theta=2^\theta,}$ or ${{\rm cf}(\lambda)>\theta}$ and ${\lambda\le\rho^\theta}$ for some ${\rho<\lambda,}$ in which case ${\lambda^\theta=\sup_{\rho<\lambda}\rho^\theta.}$ ${\Box}$

If ${I}$ is a prime ideal (i.e., the set of complements of members of an ultrafilter), then ${<_I}$ is a total quasi-order on ${\prod{\mathfrak a},}$ i.e., ${\prod{\mathfrak a}/I,}$ the collection of equivalence classes under the relation (${f=_I g}$ iff ${\{\kappa:f(\kappa)\ne g(\kappa)\}\in I}$), is linearly ordered under the relation ${[f]<_I[g]}$ iff ${\{\kappa:f(\kappa)\ge g(\kappa)\}\in I.}$ In effect, in this case ${\prod{\mathfrak a}/I}$ is just the ultrapower of ${{\mathfrak a}}$ by the ultrafilter dual to ${I.}$ By ${\mbox{\L o\'s}}$‘s lemma, this is a linearly ordered set.

This leads to the name pcf, which stands for possible cofinalities of ultraproducts of sets ${{\mathfrak a}}$ of regular cardinals.

1. A detour

Before we continue, a few words are in order on bounds for fixed points of the aleph function. For them, pcf theory does not apply directly, so different approaches are required. For example, the following extension of the ideas of Galvin-Hajnal that, combined with Theorem 6, provides bounds for powers ${\lambda^\kappa}$ for “many” ${\lambda}$ even if they are fixed points.

Theorem 10 Suppose that ${\kappa}$ is regular and ${(\lambda_i:i\le\kappa)}$ is increasing and continuous, and let ${J}$ be a normal ideal on ${\kappa.}$ Suppose that ${{\rm pp}_J(\lambda_i)\le\lambda_i^{+h(i)}}$ for all ${i.}$ Then ${{\rm pp}_J(\lambda_\kappa)\le\lambda_\kappa^{+\|h\|}.}$ ${\Box}$

Definition 11 By induction, define the classes ${C_\alpha}$ of cardinals:

1. ${C_0}$ is the class of infinite cardinals.
2. ${C_{\alpha+1}=\{\lambda\in C_\alpha:\lambda={\rm ot}(\lambda\cap C_\alpha)\}.}$
3. ${C_\lambda=\bigcap_{\alpha<\lambda}C_\alpha}$ for ${\lambda}$ limit.

For example, ${C_1}$ is the class of fixed points of the aleph sequence.

Definition 12 By induction on ${i}$ define the cardinals ${\aleph_\alpha^i(\lambda)}$ as follows:

1. ${\aleph^0_\alpha(\lambda)=\lambda^{+\alpha}.}$
2. ${\aleph_\alpha^{i+1}(\lambda)}$ is defined by induction on ${\alpha}$ as follows:
• ${\aleph^{i+1}_0(\lambda)=\lambda.}$
• ${\aleph^{i+1}_{\alpha+1}(\lambda)=\aleph^i_\gamma(\lambda),}$ where ${\gamma=\aleph^{i+1}_\alpha(\lambda)+1.}$
• If ${\delta}$ is limit, ${\aleph^{i+1}_\delta(\lambda)=\sup_{\alpha<\delta}\aleph_\alpha^{i+1}(\lambda).}$
3. If ${i}$ is limit, define ${\aleph^i_\alpha(\lambda)}$ by induction on ${\alpha:}$
• ${\aleph_0^i(\lambda)=\lambda.}$
• ${\aleph^i_{\alpha+1}(\lambda)=\sup_{j
• If ${\delta}$ is limit, ${\aleph^i_\delta(\lambda)=\sup_{\xi<\delta}\aleph^i_\xi(\lambda).}$

Note that the function ${\aleph^i_\alpha(\lambda)}$ is monotone (nonstrictly) increasing in ${i,}$ ${\alpha,}$ and ${\lambda.}$ Also, for fixed ${i,}$

$\displaystyle \{\aleph^{i+1}_\delta(\lambda):\delta\mbox{ limit}\}=\{\mu:\aleph^i_\mu(\lambda)=\mu\}.$

If ${\xi}$ is limit,

$\displaystyle \{\aleph^\xi_\alpha(\lambda):\alpha>0\}=\bigcap_{i<\xi}\{\mu:\aleph^i_\mu(\lambda)=\mu\}.$

If ${i>0,}$ ${C_i=\{\aleph_\alpha^i(\aleph_0):\alpha>0,i\mbox{ and }\alpha\mbox{ is limit}\}.}$

One then has, for example, the following bounds:

Theorem 13 Let ${{\rm nor}}$ denote the class of normal ideals.

1. If ${\zeta<\omega_1,}$ then ${{\rm pp}_{nor}(\aleph_{\omega_1}^\zeta(\beth_2(\aleph_1)))<\aleph_{\beth_2(\aleph_1)^+}^\zeta(\beth_2(\aleph_1)).}$
2. If ${\zeta<\omega_1,}$ ${\lambda}$ is the ${\omega_1}$-st member of ${C_\zeta}$ and ${\lambda>\beth_2(\aleph_1),}$ then ${{\rm pp}_{nor}(\lambda)}$ is strictly smaller than the ${\beth_2(\aleph_1)}$-st member of ${C_\zeta.}$
3. If ${{\rm cf}(\delta)=\omega_1,}$ ${\aleph_\delta>2^{\aleph_1}}$ and ${\aleph_\delta}$ is not a limit of weakly inaccessible cardinals, then there are no weakly inaccessible cardinals in ${(\aleph_\delta,{\rm pp}_{nor}(\aleph_\delta)].}$ ${\Box}$

Shelah found several additional results about bounds for powers of fixed points. However, there are inherent limitations to these results, as illustrated by the following:

Theorem 14 (Gitik) If ${{\sf GCH}}$ holds, ${\kappa}$ is a strong cardinal, ${\kappa^+<\mu,}$ and there are no inaccessibles above ${\kappa,}$ then there is a forcing extension that preserves cofinalities ${\ge\kappa,}$ and where the following hold:

1. ${\kappa}$ is the first member of ${C_\omega,}$
2. ${2^\kappa=\mu^+,}$ and
3. ${{\sf GCH}}$ holds below ${\kappa.}$ ${\Box}$

The understanding of cardinal arithmetic at fixed points is still severely more limited than at other cardinals.

2. pcf

Definition 15 Let ${{\mathfrak a}}$ be a (nonempty) set of cardinals.

1. A cardinal ${\lambda}$ is a possible cofinality of ${\prod{\mathfrak a}}$ iff there is an ultrafilter ${{\mathcal U}}$ over ${{\mathfrak a}}$ such that ${{\rm tcf}(\prod{\mathfrak a}/{\mathcal U})=\lambda.}$
2. ${{\rm pcf}({\mathfrak a})}$ is the set of possible cofinalities of ${\prod{\mathfrak a}.}$
3. ${J_{<\lambda}({\mathfrak a})=\{{\mathfrak b}\subseteq{\mathfrak a}:{\rm tcf}(\prod {\mathfrak a}/{\mathcal D})<\lambda}$ for all ultrafilters ${{\mathcal D}}$ over ${{\mathfrak a}}$ that contain ${{\mathfrak b}\}.}$

Clearly, ${J_{<\lambda}}$ is a (not necessarily proper) ideal. By considering principal ultrafilters, ${{\mathfrak a}\subseteq{\rm pcf}({\mathfrak a}).}$ One easily verifies that ${{\mathfrak a}_1\subseteq{\mathfrak a}_2}$ implies ${{\rm pcf}({\mathfrak a}_1)\subseteq{\rm pcf}({\mathfrak a}_2)}$ and that ${{\rm pcf}({\mathfrak a}\cup{\mathfrak b})={\rm pcf}({\mathfrak a})\cup{\rm pcf}({\mathfrak b}).}$

More interesting results are obtained assuming that ${{\mathfrak a}}$ is not “too large” relative to its members.

Theorem 16 Assume that ${{\mathfrak a}}$ is a set of regular cardinals and that ${|{\mathfrak a}|^+<\min({\mathfrak a}).}$ Then, for any ultrafilter ${{\mathcal D}}$ over ${{\mathfrak a},}$ ${{\rm cf}(\prod{\mathfrak a}/{\mathcal D})<\lambda}$ iff ${J_{<\lambda}({\mathfrak a})\cap{\mathcal D}\ne\emptyset.}$ ${\Box}$

It follows that (for ${{\mathfrak a}}$ as in the theorem) the sequence

$\displaystyle (J_{<\lambda}({\mathfrak a}):\lambda\in{\sf ORD})$

is increasing and continuous. Note that if ${\kappa=|\prod{\mathfrak a}|,}$ then ${{\mathcal P}({\mathfrak a})=J_{<\kappa^+}({\mathfrak a}),}$ so the displayed sequence only requires ${\lambda\le\kappa^+.}$

It also follows that in the definition of ${{\rm pp}(\lambda)}$ one can restrict oneself to prime ideals, provided that ${\lambda<\aleph_\lambda.}$

Corollary 17 With ${{\mathfrak a}}$ as in Theorem 16:

1. ${|{\rm pcf}({\mathfrak a})|\le2^{\mathfrak a}.}$
2. ${{\rm pcf}({\mathfrak a})}$ has a largest element.

The number of ultrafilters over an infinite set ${I}$ is ${2^{2^{|I|}},}$ so the bound on the size of ${{\rm pcf}}$ is nontrivial.

Proof: 1. If ${\lambda\in{\rm pcf}({\mathfrak a}),}$ then ${J_{<\lambda^+}\supsetneq J_{<\lambda}.}$ Since the sequence of ideals ${J_{<\tau}}$ is increasing and continuous, and all are contained in ${{\mathcal P}({\mathfrak a}),}$ the bound follows.

2. If ${\kappa=|\prod{\mathfrak a}|}$ then ${J_{<\kappa^+}={\mathcal P}({\mathfrak a}),}$ so there is a least ${\lambda}$ such that ${{\mathfrak a}\in J_{<\lambda^+}.}$ By continuity of the sequence of ideals ${J_{<\tau},}$ it must be the case that ${{\mathfrak a}\notin J_{<\lambda},}$ so ${\lambda}$ is regular and the largest member of ${{\rm pcf}({\mathfrak a}).}$ $\Box$

Stronger results are obtained when ${{\mathfrak a}}$ is an interval of regular cardinals:

Theorem 18 Suppose that ${{\mathfrak a}}$ is an interval of regular cardinals, without a largest element, ${{\mathfrak a}=[\min({\mathfrak a}),\sup({\mathfrak a}))\cap{\sf REG},}$ and suppose that ${|{\mathfrak a}|<\min({\mathfrak a}).}$ Then ${|\prod{\mathfrak a}|}$ is the largest element of ${{\rm pcf}({\mathfrak a})}$ and ${{\rm pcf}({\mathfrak a})}$ is also an interval of regular cardinals. ${\Box}$

It follows immediately that ${\aleph_\delta^{|\delta|}<\aleph_{(2^{|\delta|})^+}:}$ If ${2^{|\delta|}\ge\aleph_\delta}$ this is clear. Otherwise, one can take ${{\mathfrak a}=[(2^{|\delta|})^+,\aleph_\delta)\cap{\sf REG}}$ in Theorem 18.

It also follows that, for example, if ${{\mathfrak c}<\aleph_\omega,}$ then ${\aleph_\omega^{\aleph_0}}$ is regular.

By Theorem 18, ${{\rm pp}(\aleph_\omega)=\aleph_\omega^{\aleph_0}=\prod_{n>1}\aleph_n=\max({\rm pcf}(\{\aleph_n:n>1\})),}$ and therefore the bound ${{\mathfrak c}<\aleph_\omega\Rightarrow\aleph_\omega^{\aleph_0}<\aleph_{\omega_4}}$ follows from the following theorem:

Theorem 19 If ${\delta}$ is limit and ${\delta<\aleph_{\alpha+\delta},}$ then ${{\rm pp}(\aleph_{\alpha+\delta})<\aleph_{\alpha+|\delta|^{+4}}.}$ ${\Box}$

In general, for ${{\mathfrak a}}$ a “short” interval of regular cardinals, ${|{\rm pcf}({\mathfrak a})|<|{\mathfrak a}|^{+4}.}$ The main open problem in the area is whether ${|{\rm pcf}({\mathfrak a})|>|{\mathfrak a}|}$ is even possible. It has been recently shown by ${\mbox{Veli\v ckovi\'c}}$ and Er-rhaimini that Shelah’s proof cannot be improved to give the bound

$\displaystyle |{\rm pcf}({\mathfrak a})|<|{\mathfrak a}|^{+2}.$

More specifically, if ${{\mathfrak a}}$ is an interval of regular cardinals such that ${|{\mathfrak a}|<\min({\mathfrak a}),}$ then the ${{\rm pcf}}$ operator has the following properties for any ${X, Y\subseteq{\rm pcf}({\mathfrak a}):}$

1. ${X\subseteq {\rm pcf} (X ),}$ ${{\rm pcf} (X ) \cup {\rm pcf} (Y ) = {\rm pcf} (X \cup Y ),}$ ${{\rm pcf} ({\rm pcf} (X )) = {\rm pcf} (X ).}$
2. If ${\gamma\in{\rm pcf} (X ),}$ then there exists ${X'\subseteq X}$ with ${|X' | = |{\mathfrak a}|,}$ such that ${\gamma\in{\rm pcf} (X' ).}$
3. ${{\rm pcf} (X )}$ has a maximal element.
4. If ${\nu<\max{\rm pcf} ({\mathfrak a})}$ is a singular cardinal of uncountable cofinality then there exists a club ${C\subseteq\nu}$ such that ${\max{\rm pcf} ( \{\lambda^+:\lambda\in C\})=\nu^+.}$

Abstractly, properties 1–4 suffice to show that ${|{\rm pcf} ({\mathfrak a})| < |{\mathfrak a}|^{+4}.}$ Essentially, the properties allow one to identify ${{\rm pcf}}$ with a topological closure operator, and one can then argue directly about the corresponding topology. A height can be attached to these structures, and the Er-rhaimini-${\mbox{Veli\v ckovi\'c}}$ theorem is that for ${{\mathfrak a}=\{\aleph_n:n>1\}}$ it is consistent that there are topological structures satisfying properties 1–4 of height arbitrarily large below ${\aleph_3.}$

3. Kunen’s theorem revisited

Closely related to property 4 above is the following result about the existence of scales:

Theorem 20 For any singular cardinal ${\lambda}$ there is an increasing sequence ${(\lambda_i:i\in{\rm cf}(\lambda))}$ of regular cardinals cofinal in ${\lambda}$ such that ${{\rm tcf}(\prod_i\lambda_i,<_{b,{\rm cf}(\lambda)})=\lambda^+.}$ ${\Box}$

Zapletal used this result to obtain a nice proof of Kunen’s theorem that there are no embeddings ${j:V\rightarrow V.}$ His argument is as follows:

Suppose towards a contradiction that ${j:V\rightarrow M}$ is elementary and that if ${\lambda}$ is the first fixed point of ${j}$ past its critical point, then ${j[\lambda]\in M.}$ Fix ${(\lambda_i:i<\omega)}$ as in Theorem 20 with ${\lambda_0>\kappa.}$ Let ${F=(f_\alpha:\alpha<\lambda^+)}$ be strictly ${<_{b,\omega}}$-increasing and cofinal in ${\prod_i\lambda_i.}$ Then ${j(F)}$ is ${<_{b,\omega}}$-increasing and cofinal in ${j(\prod_i\lambda_i)=\prod_i j(\lambda_i)}$ in the sense of ${M.}$ Since ${j[\lambda]}$ is cofinal in ${\lambda,}$ then in ${M,}$ ${j[F]}$ is also ${<_{b,\omega}}$-increasing and cofinal in ${\prod_i j(\lambda_i).}$

Now define a function ${g}$ with domain ${\omega}$ by ${g(i)=\sup(j[\lambda]\cap j(\lambda_i))=\sup(j[\lambda_i]).}$ Note that since ${j(\lambda_i)}$ is regular in ${M}$ and strictly larger than ${\lambda_i,}$ then ${g(i)\in j(\lambda_i),}$ so ${g\in\prod_i j(\lambda_i).}$

However, ${j(f_\alpha)(i)=j(f_\alpha(i))<\sup (j[\lambda_i])=g(i)}$ for all ${i<\omega}$ and all ${\alpha<\lambda,}$ so ${j[F]}$ is not cofinal in ${\prod_ij(\lambda_i)}$ after all, since ${g}$ bounds it. Contradiction.

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### 7 Responses to 580 -Cardinal arithmetic (12)

1. Theorem 5 is incorrect as stated. I’ll post a correct version soon. Thanks to A. Rinot for pointing out the problem.

2. The post has been updated and debugged. Once again, thanks to A. Rinot for spotting the problem.

3. […] gives us another proof of Kunen’s inconsistency Theorem 13 from lectures II.10 and II.12. Corollary 9 (Kunen) If is elementary, […]

4. […] from Theorem 20 in lecture II.12 that for every singular cardinal there is an increasing sequence of regular cardinals cofinal in […]

5. […] that there are no embeddings All the known proofs of this result (see for example lectures II.10, II.12, and III.3) use the axiom of choice in an essential way. Theorem 27 (Sargsyan) In assume there […]

6. […] that there are no embeddings All the known proofs of this result (see for example lectures II.10, II.12, and III.3) use the axiom of choice in an essential way. Theorem 27 (Sargsyan) In assume there […]

7. […] This gives us another proof of Kunen’s inconsistency Theorem 13 from lectures II.10 and II.12. […]