Caveat Lector

May 25, 2011 at 00:02 | Posted in Uncategorized | Leave a comment

I’ve had a chance to look over some of the posts for the second project, and I’ve been trying to fix typos and other infelicities, where by “infelicities” I mean “stupid things I say because I type without proofreading…”

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Claim 3.3A, or, More Elementary Submodels (iii)

May 24, 2011 at 10:47 | Posted in Uncategorized | Leave a comment

Here we conclude a trio of posts dealing with elementary submodels and Skolem hulls. My goal in this post is to prove a slight strengthening of Observation 3.3A on page 379 of Cardinal Arithmetic, a result which will fill a critical role in the proof of the main theorem. Notice that what we are doing is just putting the two previous posts together into a single result.

Lemma 1 Suppose {\mu} is a singular cardinal, let {\mathfrak{A}} be as usual, and let {M} be an elementary submodel such that {\mu\in M}. If there are objects {\eta} and {H} such that

  • {\eta} is a cardinal below {\mu},
  • {\eta=\sup(M\cap\eta)}
  • {H\subseteq M\cap\prod((\eta,\mu)\cap{\sf Reg})}, and
  • for every {\kappa\in M\cap\prod ((\eta,\mu)\cap{\sf Reg})},

    \displaystyle  \sup\{f(\kappa):f\in H\}=\sup(M\cap\kappa), \ \ \ \ \ (1)

    then

    \displaystyle  M\cap\mu\subseteq N:={\rm Sk}_{\mathfrak{A}}(\eta\cup H\cup\{\mu\}). \ \ \ \ \ (2)

Proof:

We see that the assumptions of Proposition 2 from yesterday’s post are satisfied if we define {p} to be {H\cup\{\mu\}}, and therefore the assumptions of Theorem 1 from the May 18, 2011 post are satisfied. If {M\cap\mu} is not a subset of {N}, then there is a regular {\kappa\in M\cap N\cap\mu} such that

\displaystyle  \sup(N\cap\kappa)<\sup(M\cap\kappa). \ \ \ \ \ (3)

But then there is an {f\in H} such that

\displaystyle  \sup(N\cap\kappa)<f(\kappa)<\kappa, \ \ \ \ \ (4)

which is a contradiction as both {f} and {\kappa} are in {N}. \Box

More elementary submodels (II)

May 23, 2011 at 16:55 | Posted in Uncategorized | Leave a comment

In this post, I want to give a more precise description of “Skolem hull arguments”, as such arguments are critical ingredients in many of the proofs we are looking at in this blog. I’ve done some searching, and I think I like the way Holst, Steffens, and Weitz present things in their book “Introduction to Cardinal Arithmetic” (referred to in the sequel as [HSW]), as they actually fill in several details that are usually glossed over and we’ll shortly be in need of this level of precision. I’m just going to sketch a little material from section 4.4 of their book, mainly to fix notation.

Our convention is that {\mathfrak{A}} denotes a structure of the form {\langle H(\chi), \in , <_\chi\rangle}, where {<_\chi} is a well-ordering of {H(\chi)}. We’ll denote the usual language of set theory (i.e., one binary relation symbol standing for {\in}) by {L}, and let {L^*} be {L} together with another binary relation symbol standing for {<_\chi}.

Given an {L^*} formula {\phi(x_0,\dots, x_n)} in which {x_0} occurs freely, let {f_\phi} be the function from {H(\chi)^n\rightarrow H(\chi)} which sends each {n}-tuple {\langle a_1,\dots,a_n\rangle} to the {<_\chi}-least element {a\in H(\chi)} for which

\displaystyle  H(\chi)\models\phi(a, a_1,\dots, a_n), \ \ \ \ \ (1)

if there is such an {a}, and which returns the value {0} if there is no such {a}. Thus, {f_\phi} is a Skolem function for {\phi}.

We usually think of the Skolem hull of {X\subseteq H(\chi)} as being the closure of {X} under all Skolem functions, but the fact that our Skolem functions are definable in {L^*} gives us the following:

Lemma 1 (Lemma 4.4.2 of [HSW]) If {X\subseteq H(\chi)}, then

\displaystyle  {\rm Sk}_{\mathfrak{A}}(X)=\bigcup\{ f_\phi[X]:\phi\text{ a formula in }L^*\}. \ \ \ \ \ (2)

The above will help make some of the arguments we present less technical than they might have been. Of course we will still be sloppy and refer to “elementary submodels of {H(\chi)}”. One can take this as either “officially” referring to elementary substructures of {\mathfrak{A}}, or referring to the {L}-reducts of such structures.

For an easy example of a Skolem hull argument, let’s look again at the preceding post. Back then, we were considering elementary submodels {M} and {N} of {H(\chi)}, where {\chi} was a regular cardinal much larger than some fixed singular cardinal {\mu}. The theorem proved there needed two assumptions:

\displaystyle  \mu\in M\cap N, \ \ \ \ \ (3)

and

\displaystyle   (\forall\alpha\in N\cap\mu)(\exists A\in M)(\alpha\in A\wedge A\subseteq N), \ \ \ \ \ (4)

and now we can prove the following proposition:

Proposition 2 The two assumptions above are satisfied if

\displaystyle  N={\rm Sk}_{\mathfrak{A}}(\eta\cup p) \ \ \ \ \ (5)

where

  1. {\mu\in p\subseteq M},
  2. {\eta} is a cardinal below {\mu}, and
  3. {\eta=\sup(M\cap\eta)}.

Proof: Clearly only (4) is an issue, so assume {\alpha\in N\cap \mu}. We know there is an {L^*} formula {\phi}, ordinals {\alpha_0,\dots,\alpha_{m-1}} in {\mu}, and {a_0,\dots, a_{n-1}} in {p} such that

\displaystyle  \alpha=f_\phi(\alpha_0,\dots,\alpha_{m-1},a_0,\dots,a_{n-1}). \ \ \ \ \ (6)

Fix {\zeta\in M\cap\eta} greater than {\max(\alpha_0,\dots,\alpha_{m-1})}, and define

\displaystyle  A:=\{\beta<\mu:(\exists\beta_0<\zeta)\dots(\exists\beta_{m-1}<\zeta)\phi(\beta,\beta_0,\dots,\beta_{m-1},a_0,\dots, a_{n-1})\}. \ \ \ \ \ (7)

The set {A} definable in {\mathfrak{A}} with parameters from {M}, hence {A\in M}. Notice that {A} lies in {N} as well, as {\zeta\in N} and {p\subseteq N}. Since {\zeta} (and hence {\zeta^n}) is a subset of {N}, it follows that {A\subseteq N}. Clearly {\alpha\in A}, too, and so we have what we want. \Box

******
Edit 5/27/2011 I left a piece of formula out of (7) —- there should be a “such that beta is the least such that…” in the definition of A. I’ll fix it when I get time.

More elementary submodels

May 18, 2011 at 14:09 | Posted in Uncategorized | Leave a comment

This post should be considered a tentative one. What I’m trying to do is isolate a few facts that can be used to prove the various “elementary submodel results” scattered throughout the last chapter of Cardinal Arithmetic. I’ve already gone through several iterations of this, but the following result looks “stable” so I thought I’d put it up here.

Theorem 1 Suppose {\mu} is a singular cardinal, and let {\chi} be a sufficiently large regular cardinal. Suppose {M} and {N} are elementary submodels of {\langle H(\chi), \in, <_\chi\rangle} such that

  1. {\mu\in M\cap N}, and
  2. {(\forall\alpha\in N\cap\mu)(\exists A\in M)(\alpha\in A\wedge A\subseteq N)}.

If {M\cap\mu\nsubseteq N}, then there is a regular cardinal {\kappa\in M\cap N\cap\mu} such that

\displaystyle  \sup(N\cap\kappa)<\sup(M\cap\kappa). \ \ \ \ \ (1)

Proof: Assuming {M\cap\mu\nsubseteq N}, we define

\displaystyle  \gamma(*):=\min(M\cap\mu\setminus N), \ \ \ \ \ (2)

and

\displaystyle  \beta(*):=\min(N\cap{\sf On}\setminus\gamma(*)). \ \ \ \ \ (3)

Note the following facts:

\displaystyle  \gamma(*)<\beta(*)\leq\mu, \ \ \ \ \ (4)

and

\displaystyle   N\cap [\gamma(*),\beta(*))=\emptyset. \ \ \ \ \ (5)

Note that {\beta(*)} cannot be a successor ordinal: if it were, then its predecessor would witness the failure of (5). Since {\gamma(*)+1\in M}, we conclude

\displaystyle  \sup(N\cap\beta(*))\leq\gamma(*)<\gamma(*)+1\leq\sup(M\cap\beta(*)), \ \ \ \ \ (6)

and so our proof will be complete if we can show that {\beta(*)} is in fact a regular cardinal.

Our proof hinges on the fact that {\beta(*)} is an element of {M} as well. Why is this? By our assumptions, there is a set {A\in M} such that

\displaystyle  \beta(*)\in A\text{ and }A\subseteq N. \ \ \ \ \ (7)

Since {N\cap [\gamma(*),\beta(*))=\emptyset}, we conclude

\displaystyle  \beta(*)=\min(A\setminus\gamma(*)), \ \ \ \ \ (8)

and hence {\beta(*)} is definable from parameters available inside the model {M}.

Thus, we have {\beta(*)\in M\cap N}. If {\beta(*)} is not a regular cardinal, then

\displaystyle  {\rm cf}(\beta(*))\in M\cap N\cap\beta(*), \ \ \ \ \ (9)

and therefore

\displaystyle  {\rm cf}(\beta(*))\leq\gamma(*). \ \ \ \ \ (10)

Given our definition of {\gamma(*)}, we see

\displaystyle   M\cap{\rm cf}(\beta(*))\subseteq N. \ \ \ \ \ (11)

We can find {f\in M\cap N} such that {f} maps {{\rm cf}(\beta(*))} onto a cofinal subset of {\beta(*)}. Since {\gamma(*)\in M}, it follows that there is an {\alpha\in M\cap {\rm cf}(\beta(*))} such that

\displaystyle  \gamma(*)<f(\alpha)<\beta(*). \ \ \ \ \ (12)

But we know {f\in N}, and {\alpha\in N} by (11), and therefore

\displaystyle  f(\alpha)\in N\cap [\gamma(*),\beta(*)), \ \ \ \ \ (13)

contradicting (5)

\Box

Strategy and assumptions

May 10, 2011 at 16:21 | Posted in Uncategorized | Leave a comment

In this post, we’ll lay out the notation and strategy for the proof.

Let us assume {\sigma}, {\theta}, {\mu}, and {\lambda} are cardinals with

\displaystyle  \aleph_0<\sigma\leq{\rm cf}(\mu)<\theta\leq\mu, \ \ \ \ \ (1)

such that

\displaystyle  {\rm tcf}(\prod\mathfrak{a}, <_I)<\lambda \ \ \ \ \ (2)

whenever {\mathfrak{a}} and {I} satisfy

  • {\mathfrak{a}} is a progressive set of regular cardinals with {\sup(\mathfrak{a})=\mu} and {|\mathfrak{a}|<\theta},
  • {I} is a {\sigma}-complete ideal on {\mathfrak{a}} extending the bounded ideal, and
  • the structure {(\prod\mathfrak{a}, <_I)} has a true cofinality.

To save ourselves some time later, let us agree to call such {\mathfrak{a}} and {I} relevant.

Now let {\chi} be a sufficiently large regular cardinal, and let {M} be an elementary submodel of {\langle H(\chi),\in, <_\chi\rangle} satisfying

  • {|M|<\lambda},
  • {\{\sigma,\theta,\mu, \lambda\}\in M}, and
  • {M\cap\lambda} is an ordinal of cofinality greater than {\theta}.

Our goal is to show that the collection

\displaystyle  \mathcal{P}:= M\cap [\mu]^{<\mu} \ \ \ \ \ (3)

witnesses {{\rm cov}(\mu,\mu,\theta,\sigma)\leq |M|}, i.e., that every set in {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.

First steps in the hard direction

May 10, 2011 at 16:01 | Posted in Uncategorized | Leave a comment

Now what about the “hard direction”? Given cardinals {\sigma}, {\theta}, and {\mu} such that {\sigma} is regular and {\aleph_0<\sigma\leq{\rm cf}(\mu)<\theta\leq\mu}, our aim is to prove

\displaystyle  {\rm cov}(\mu,\mu,\theta,\sigma)\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (1)

and moreover, if {{\rm cov}(\mu,\mu,\theta,\sigma)} is regular, then there exist {\mathfrak{a}} and {I} such that

  1. {\mathfrak{a}} is a progressive set of regular cardinals with {\sup(\mathfrak{a})=\mu} and {|\mathfrak{a}|<\theta},
  2. {I} is a {\sigma}-complete ideal on {\mathfrak{a}} extending the bounded ideal,
  3. the structure {(\prod\mathfrak{a}, <_I)} has a true cofinality, and
  4. {{\rm cov}(\mu, \mu, \theta, \sigma)\leq {\rm tcf}(\prod\mathfrak{a}, <_I)}.

Notice that by the preceding post, condition 4 is equivalent to the statement

\displaystyle  {\rm cov}(\mu, \mu, \theta,\sigma)={\rm tcf}(\prod\mathfrak{a}, <_I), \ \ \ \ \ (2)

and so the equality

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)={\rm cov}(\mu,\mu,\theta,\sigma) \ \ \ \ \ (3)

holds (at least in the case where {\aleph_0<\sigma}) in a strong sense. Note that the underlying issue here is that the pp number is defined as the supremum of a certain set of regular cardinals; we will show that if this supremum itself is a regular cardinal (and hence weakly inaccessible), then it actually occurs as the true cofinality of {(\prod\mathfrak{a}, <_I)} for some relevant {\mathfrak{a}} and {I}.

We’ll do this by proving the following:

Theorem 1 Suppose {\lambda} is a regular cardinal such that {{\rm tcf}(\prod\mathfrak{a}, <_I)<\lambda} whenever {\mathfrak{a}} and {I} satisfy conditions 1, 2, and 3 above. Then {{\rm cov}(\mu,\mu,\theta,\sigma)<\lambda} as well.

The usual caveats apply here: it is closer to the truth to say that I will attempt to prove the above theorem. We’ll see what happens…

The easy direction

May 10, 2011 at 15:44 | Posted in Uncategorized | Leave a comment

In this post, we’ll prove a more nuanced version of the result we discussed in our January 31 post. This is the “easy direction” of the theorem we’re aiming to prove, and the argument goes through without any special assumptions on the cofinality of the singular cardinal {\mu}.

Theorem 1 Suppose {\sigma}, {\theta}, and {\mu} are cardinals satisfying {\sigma\leq{\rm cf}(\mu)<\theta\leq\mu}. Then

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)\leq{\rm cov}(\mu,\mu, \theta,\sigma). \ \ \ \ \ (1)

Proof: Suppose {\mathfrak{a}\subseteq\mu} is a progressive set of regular cardinals such that {|\mathfrak{a}|<\theta} and {\sup(\mathfrak{a})=\mu}. Further assume that {I} is a {\sigma}-complete ideal on {\mathfrak{a}} extending the bounded ideal such that {\prod\mathfrak{a}/I} has true cofinality, and let

\displaystyle  \kappa:={\rm tcf}(\prod\mathfrak{a}, <_I). \ \ \ \ \ (2)

We must show

\displaystyle  \kappa\leq{\rm cov}(\mu, \mu, \theta,\sigma). \ \ \ \ \ (3)

Suppose this fails, and let {\mathcal{P}} witness {{\rm cov}(\mu,\mu,\theta,\sigma)<\kappa}. By our choice of {\kappa}, there is a sequence {\langle f_\alpha:\alpha<\kappa\rangle} such that

  • each {f_\alpha} is in {\prod\mathfrak{a}},
  • {\alpha<\beta\Longrightarrow f_\alpha<_I f_\beta}, and
  • if {g\in\prod\mathfrak{a}}, then {g<_I f_\alpha} for some {\alpha<\kappa}.

Since {|\mathfrak{a}|<\theta}, it follows that for each {\alpha<\kappa} the range of {f_\alpha} is covered by a union of fewer than {\sigma} sets from {\mathcal{P}}. Since {I} is {\sigma}-complete, we conclude that for each {\alpha<\kappa} there is a set {A_\alpha\in\mathcal{P}} such that

\displaystyle  \{\eta\in\mathfrak{a}:f_\alpha(\eta)\in A_\alpha\}\notin I. \ \ \ \ \ (4)

Now {\kappa} is regular and {|\mathcal{P}|<\kappa}, so we can assume that there is a single set {A\in\mathcal{P}} with the property that

\displaystyle  B_\alpha:=\{\eta\in\mathfrak{a}:f_\alpha(\eta)\in A\}\notin I \ \ \ \ \ (5)

for each {\alpha<\kappa}.

Now define a function {g\in\prod\mathfrak{a}} as follows:

\displaystyle  g(\eta)= \begin{cases} \sup(A\cap\eta) &\text{if this is less than }\eta,\\ 0 &\text{otherwise.} \end{cases} \ \ \ \ \ (6)

Since {|A|<\theta\leq\mu}, we know that {A\cap\eta} is bounded in {\eta} for all sufficiently large {\eta\in\mathfrak{a}}. Since {I} contains the bounded subsets of {\mathfrak{a}}, it follows that for each {\alpha<\kappa},

\displaystyle  \{\eta\in\mathfrak{a}: f_\alpha(\eta)\leq g(\eta)\}\in I^+. \ \ \ \ \ (7)

But this is a contradiction, as there must exist an {\alpha<\kappa} such that {g<_I f_\alpha}. \Box

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