Tuesday, December 30, 2014

Good and bad terminology in mathematics

Happy holidays! I hope everyone is having a restful vacation. Here's a post which is just for a bit of fun---don't take it too seriously!

A lot of people complain about bad terminology in mathematics. I don't really consider myself to be one of those people, but I have to admit they have a point. The choice of a word we use to represent a concept greatly influences the way we read and perceive it. Here are examples of terminology I like and dislike, with some whimsical alternatives for fun. It turned out to be much harder than I expected to come up with good terminology. Leave a comment if you can think of something to add to these lists.

  • Adjoint: This word seems to only exist in mathematics. It captures a general pattern that appears frequently in different contexts; several formally related concepts share this terminology. Appropriately, it suggests pairing or duality.
  • Club set: Short for "closed and unbounded". This neat portmanteau helps emphasize that the conjunction of the two properties is somehow more the sum of its parts.
  • Compactness: This is such a carefully nuanced word for a subtle and fundamental property which occurs in many different contexts in mathematics, although I think it can take some getting used to. It really captures the idea of some kind of finitary character, or a strong notion of "smallness".
  • Mouse, weasel: Inner model theory words that give a mischievous flavor to a difficult subject.
  • Mixing: from ergodic theory. It's a gerund, which helps describe this dynamical situation well.
  • Rational numbers: This is a nice play on words. It makes $\mathbb{Q}$ feel very concrete and friendly and is probably derived from the representation of these numbers as ratios of integers. 
  • Supercompact cardinal: sometimes terminology doesn't totally make sense, it just sounds cool. I have to admit that hearing this word early made me curious about set theory.

  • Amenable group: This seems to be a pun that only makes sense with British pronunciations. According to Wikipedia, the original name given by von Neumann was "messbar", or "measurable" in English. Of course, this name is even worse. Alternatives: Maybe something like "well-measured" would be better.
  • Antichain: It's actually a catchy word, if only it didn't have two closely related but nonequivalent meanings. In order theory people use it to mean a collection of pairwise incomparable elements of a poset, while set theorists would say it is a collection of pairwise incompatible elements. Let the order theory people take back the outdated term "Sperner system".
  • Cardinal collapsing: This term from set theory is really convenient and probably no one else in the world has a problem with it. My gripe is that it's somewhat inaccurate and easily replaced by simpler terms. "Collapsing" a cardinal refers to a forcing extension where a particular ordinal is no longer a cardinal, in other words, its cardinality is decreased (as measured by ordinals) in the extension. The word "collapse" is a bit overloaded (it seems better suited to describe things like Mostowski collapse). Alternatives: "decardinalizing", "cardinality decreasing".
  • Commutative diagram: To commute just means to go from one place to another. This got twisted into the property that $ab=ba$; I guess I can kind of see that they are moving past each other. So we can also use this for functions $f,g:X\rightarrow X$ to say $fg=gf$ (a special kind of commutative square). But what about commutative triangles, some functions that satisfy $f=gh$? Alternatives: "freely composing", or maybe something made-up and descriptive like "ambicompositional".
  • Countable chain condition (c.c.c.): This one from set theory is unpopular among some people I know, since it's a property of posets that means there are no uncountable antichains (and I don't mean Sperner systems here). I saw one paper (Abraham--Shelah: Forcing closed unbounded sets) which called this the c.a.c., for countable antichain condition, which seems like a reasonable alternative. I recall that there's actually a good historical reason for this terminology, though I'm not sure exactly what that was.
  • Maths: OK, so it's not really in the spirit of the other things, and maybe its just my American crudeness, but this sounds totally wrong to me.
  • One-to-one: I thought it was OK until someone pointed out to me that it sounds like each member of the domain maps to a unique member of the codomain (what some people call single-valued). This has been the source of some confusion in classes I've TA'ed. Alternatives: just use the already existing technical-sounding words "injective" and "surjective."
  • Proper forcing: It's just a bit too vague and authoritarian for me. Alternatives: Maybe something like "internally generic forcing".

Sunday, December 14, 2014

Spencinar flashback: Approachability and stationary reflection at $\aleph_\omega$

The Spencinar has been continuing as usual since the Thanksgiving break. Two weeks ago, Spencer talked about diamond principles. I'm working on these notes still. Last week, I talked about precipitousness of the nonstationary ideal (the subject of the learning project #1).

I'm about to retire this old seminar notebook, so I thought I would record here a Spencinar that I gave over the summer about the approachability property. This will also provide some useful background for future posts.

This material comes mostly from Eisworth's friendly handbook chapter, where I was originally inspired to pursue the line of research that I'm on. I'm going to focus on the following result of Shelah's (interpreted through Eisworth's chapter):

Main Theorem: If $\aleph_\omega$ is a strong limit, then $\rm{Refl}(\aleph_{\omega+1})$ implies the approachability property at $\aleph_\omega$.

This result is unexpected since approachability is a rather weak "non-reflection" principle. It's interesting since Magidor's model of $\rm{Refl}(\aleph_{\omega+1})$ satisfies approachability property at $\aleph_\omega$, and in fact uses this to show that stationarity is preserved after forcing. It's also interesting since the proof seems to use special properties of $\aleph_\omega$, giving an example of the phenomenon that this singular cardinal in particular seems to have a lot of ZFC structure. This is nice to understand, because there is an important open question asking whether the failure of singular cardinals hypothesis at $\aleph_\omega$ implies approachability (and this is consistently false for larger singular cardinals).

I. $\mathrm{AP}_\mu$: review.

Shelah defined the ideal $I[\lambda]$ for a regular cardinal $\lambda$ by:

$S\subseteq I[\lambda]$ iff there is

  1. a sequence $\langle a_\alpha:\alpha<\lambda\rangle$ of bounded subsets of $\lambda$ and
  2. $C\subseteq \lambda$ club,

 so that for every $\delta\in S\cap C$, there is $A_\delta\subseteq \delta$ unbounded of order-type $\rm{cf}(\delta)$ so that

  • $\{A\cap \beta:\beta<\delta\}\subseteq \{a_\beta:\beta<\delta\}$ (or in words, all initial segments of $A_\delta$ are enumerated in $\bar{a}$ at a stage before $\delta$.
Such $\delta$ is said to be approachable with respect to $\bar{a}$. It can be shown (cf. Eisworth's handbook chapter) that $I[\lambda]$ is a normal ideal on $\lambda$.

For $\mu$ a singular cardinal, the approachability property $\rm{AP}_\mu$ is the statement $\mu^+\in I[\mu^+]$. (Some other authors denote this by $\rm{AP}_{\mu^+}$. Restating the definitions, this means there is $\bar{a}$ and a club of $\delta<\mu^+$ of points which are approachable with respect to $\bar{a}$.

2. Elementary substructures

We will define a $\lambda$-approximating sequence to be a $\subseteq$-increasing sequence $\mathcal{M}=\langle M_\alpha:\alpha<\lambda\rangle$ of elementary substructures of $(H(\theta); \in, \triangleleft)$ for sufficiently large regular $\theta$ so that $\lambda\in M_0$ and for all $\alpha$, $|M_\alpha|<\lambda$ and $M_\alpha\cap \lambda\in \lambda$. Most importantly, we require that $\langle M_\alpha:\alpha\le \beta\rangle\in M_{\beta+1}$. This is a slight strengthening of the concept of "IA sequence" which we saw in the learning project.

If $\mathcal{M}$ is a  $\lambda$-approximating sequence, then define $S[\mathcal{M}]$ to be the set of $\delta<\lambda$ such that $M_\delta\cap\lambda=\delta$, there is a cofinal $a\subseteq \delta$ of order-type $\rm{cf}(\delta)$ such that every inital segment of $a$ is in $M_\delta$.

This gives a useful characterization of approachability. $I[\lambda]$ is just ideal generated by nonstationary sets together with sets of the form $S[\mathcal{M}]$. For a given $\bar{a}$, let $\mathcal{M}$ be a $\lambda$-approximating sequence with $\bar{a}\in M_0$ and $E\subset\lambda$ be club so that $M_\alpha\cap\lambda=\alpha$ for all $\alpha\in E$. In this club, the set of points approachable with respect to $\bar{a}$ is contained in $S[\mathcal{M}]\cap E$.

Conversely, given a sequence $\mathcal{M}$, let $\bar{a}$ be the sequence of bounded subsets of $\lambda$ in $\bigcup_\alpha M_\alpha$, and $E$ be the club of $\delta$ so that $\{a_\alpha:\alpha<\delta\}$ is an enumeration of the bounded subsets of $M_\delta$.

3. Colorings

We can also rephrase notions of approachability in terms of colorings, given some extra assumptions. Here $S^\lambda_\kappa$ denotes the set of points of cofinality $\kappa$ below $\lambda$.

Theorem 1: Suppose $\kappa<\lambda$ be regular with $2^{<\kappa}<\lambda$. If $d:[\lambda]^2\rightarrow \omega$ and $\mathcal{M}$ is a $\lambda$-approximating sequence containing $\{\kappa, d\}$, then for every $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, there is a cofinal $H\subseteq \delta$ homogeneous for d.

Proof: If $\delta\in S[\mathcal{M}]\cap S^\lambda_\kappa$, let $\{\alpha_i:i<\kappa\}$ cofinal in $\delta$ such that $\{a_i:i<\zeta\}\in M_\delta$ for all $\zeta<\kappa$. By induction on $i<\kappa$, we can define $\epsilon_i,f_i$ so that:

  1. $\epsilon_0=\alpha_0$.
  2.  $f_i:i\rightarrow \theta$, $f_i(j)=d(\epsilon_j,\delta)$.
  3. $\epsilon_i$ is the least $\alpha$ so that for all $j<i$, $\alpha>\alpha_j,\epsilon_j$ and $d(\epsilon_j,\alpha)=d(\epsilon_j,\delta)$ for all $j<i$, if such exists. The construction ends if no such exists.
This last condition says that the function $d(\epsilon_i,\epsilon_j)$ depends only on $i$, where $i<j$, and uses $\delta$ as a "reference point". Let $i^*$ be the length of the construction.

We show the construction goes up to $\delta$. By part of the definition of $S[\mathcal{M}]$, $M_\delta\cap\lambda=\delta$. Since $\kappa\in M_\delta$, $M_\delta$ must be $<\kappa$-closed, so it contains $f_i$ for all $i<i^*$ (this uses $2^{<\kappa}<\lambda$). This means the construction above can be done in $M_\delta$ and therefore all $\epsilon_i<\delta$, $\delta$ witnesses condition in 3.

The proof concludes by finding $\xi$ so that $\{i<\kappa:d(\epsilon_i,\delta)=\xi\}$ is unbounded, giving a homogeneous set. $\Box$

This motivates yet another conception of "approachability".

Definition: For $d:[\lambda]^2\rightarrow\chi$ (some $\chi$), define $S(d)$ to be the set of $\delta<\lambda$ for which there is a cofinal $H_\delta\subseteq \delta$ homogeneous for $d$, and $S^*(d)=\lambda-S(d)$.

So $S(d)$ is club for every $d$ if the approachability property holds at the predecessor of $\lambda$ (under the hypotheses of Theorem 1).

Actually, under those hypotheses, the approachability ideal is generated by a single set over the nonstationary ideal. We can see this result in a way by using colorings. It turns out there is a certain class of colorings whose members have "all of the complexity" of the general case.

Definition: $d:[\lambda]^2\rightarrow\omega$ is a coloring as above with $\lambda=\mu^+$. We say $d$ is normal if $\{\beta<\alpha:d(\beta,\alpha)<i\}|<\mu$ for all $i<\rm{cf}(\mu)$. We say $d$ is transitive if $d(\alpha,\gamma)\le \mathrm{max}\{d(\alpha,\beta),d(\beta,\gamma)\}$ for all $\alpha<\beta<\gamma<\lambda$.

Normal transitive colorings are not difficult (but a little tricky) to construct. We'll just take this as an unproven fact (this construction was used, for example, in John's Spencinar talk about Martin's Maximum).

Proposition: If $\mu$ is strong limit and $d$ is a normal, transitive coloring of $\mu^+$, then there is $\bar{a}$ so that almost every point of $S(d)$ is approachable with respect to $\bar{a}$.

Proof: Let $\langle a_\xi:\xi<\lambda\rangle$ enumerate all subsets of the form $\{\beta<\alpha:d(\beta,\alpha)<i\}$ in some order. This is fine since $\lambda^{<\lambda}=\lambda$ ($\mu$ strong limit). Then for any $\delta\in S(d)$ of uncountable cofinality there exists a cofinal $H_\delta\subseteq\delta$ homogeneous for $d$. Actually, $d(\beta,\delta)$ is bounded in $\omega$: by transitivity, if $\beta_1<\beta_2<\delta$, then $d(\beta_1,\delta)\le \max\{i,d(\beta_2,\delta)\}$, and since $\delta$ has uncountable cofinality and $d(\beta,\delta)$ is increasing in $\beta\in H_\delta$ (if above $i$), then $d(\beta,\delta)$ is bounded. So almost every point of $S(d)$ is approachable. $\Box$

So the proposition gives a really nice characterization of approachability property under the hypotheses of Theorem 1: $\mathrm{AP}_\mu$ holds iff there exists (equivalently for all) normal transitive colorings, $S(d)$ contains a club of points of uncountable cofinality.

4. Proof of Main Theorem

Now we start working towards the proof of the theorem. The first result is that $S^*(d)$ can only reflect in itself.

Lemma 2: If $S^*(d)\cap \alpha$ is stationary in $\alpha$, then $\alpha\in S^*(d)$.

Proof: Otherwise $\alpha\in S(d)$, so let $H\subset \alpha$ be cofinal in $\alpha$ and homogeneous for $d$. Then every limit point of $H$ must be in $S(d)$ by the definition of $S(d)$, so $S^*(d)$ is disjoint from a club in $\alpha$. $\Box$

So if the approachability property fails, then there is a coloring $d$ (namely, any normal transitive one) for which $S^*(d)\cap \mathrm{cof}(>\omega)$ is stationary. Take $n<\omega$ such that $S=S^*(d)\cap S^{\aleph_{\omega+1}}_{\aleph_{n+1}}$ is stationary.

The next claim is the crucial fact giving a non-reflecting stationary set.

Claim: $S$ cannot reflect at points of cofinality $>2^{\aleph_n}$.

Proof of Claim: Suppose $S$ reflects at $\tau$.  If $\tau>2^{\aleph_n}$, then $\tau^{<\aleph_{n+1}}=\tau$. So we can construct a $\lambda$-approximating sequence containing all the relevant parameters which is $<\aleph_{n+1}$ closed. Then for any $\delta\in S^\tau_{\aleph_{n+1}}$ with $M_\delta=\delta$, any $A\subseteq \delta$ of order-type ${\aleph_{n+1}}$ has all of its initial segments in $M_\delta$. This shows that $S^\tau_{\aleph_{n+1}}\in I[\tau]$. This is impossible by Theorem 1, which says almost all points below $\tau$ are in $S(d)$, so $S$ cannot reflect at $\tau$. $\Box$

Now fix $k<\omega$ so that $2^{\aleph_n}=\aleph_{n+k}$. Such exists by $\mu$ strong limit. Define $S_0=S$, and $S_{i+1}$ to be the reflection points of $S_i$. Observe that if $\delta\in S_i$, then $\mathrm{cf}(\delta)\ge \aleph_{n+1+i}$, since the set of points below an ordinal with cofinality less than that ordinal is club. This implies that $S_k$ is empty, so let $i^*<\omega$ be maximal so that $S_{i^*}$ is stationary. Then $S_{i^*}$ does not reflect stationarily often. Hence we have proven the main theorem.

5. Normal scales?

Let's try to apply some of these ideas. This section requires a basic knowledge of pcf theory. Let $\langle f_\alpha:\alpha<\mu^+\rangle$ be a scale.

The "difference function" of the scale is a transitive function $d:[\mu^+]^2\rightarrow\omega$ defined so that $d(\alpha,\beta)$ for $\alpha<\beta$ is the supremum of all $n$ so that $f_\alpha(n)\ge f_\beta(n)$. In light of the previous explorations on approachability, it is natural to ask if it can ever be normal.

Proposition: Normal scales don't exist.

Proof: Let $\vec{f}$ be a scale on $\langle \mu_i:i<\omega\rangle$. Pick $\gamma_1<\mu_1$ so that $B_1=\{\alpha:f_\alpha(1)<\gamma_1\}$ has size $\kappa^+$. Let $A_0$ be a set of $\mu_0$ many $\alpha<\mu^+$ in $B_1$.

If $n>0$, then pick $\gamma_{n+1}$ s.t. $\gamma_{n+1}>\sup_{\alpha\in A_{n-1}} f_\alpha(n+1)$ and $B_{n+1}=\{\alpha\in B_n: f_\alpha(n+1)<\gamma_{n+1}\}$ has size $\mu^+$. Let $A_n$ be a set of $\mu_{n}$ many from $B_{n+1}$.

Then $\bigcup_n A_n$ has size $\kappa$ and is bounded by $n\mapsto \gamma_n$.